This course develops the structure theory of compact Lie groups, moving from analytic foundations to the full classification and representation-theoretic picture. It asks how compactness simplifies the geometry and algebra of Lie groups, and how one can recover global information from invariant integration, conjugacy, and the action of a group on its own [Lie algebra](/page/Lie%20Algebra). The central goal is to understand compact Lie groups through the interplay of symmetry, harmonic analysis, and combinatorial data such as roots, weights, and lattices. The early chapters build the analytic and geometric tools: Haar measure and averaging on compact groups, maximal tori, conjugacy theorems, and the Weyl [group action](/page/Group%20Action). From there the course passes to root systems and the adjoint representation, which encode the internal geometry of the group, and then to lattices and weight data, which organize its representations. Highest weight theory, Peter-Weyl theory, and character orthogonality provide the representation-theoretic backbone, while the Weyl integration and character formulas translate abstract structure into computable formulas. The final chapter ties these ideas to the compact classical families, showing how the general theory manifests in familiar examples such as unitary, orthogonal, and symplectic groups. # Introduction This introductory chapter fixes the viewpoint of the course before the structural theory begins. The first course on Lie groups builds the language of manifolds, Lie algebras, homomorphisms, actions, and the exponential map. Here the emphasis shifts from local differential structure to global structure: for compact connected Lie groups, much of the group can be read from a maximal torus, provided we remember the finite symmetry group acting on it. The main examples throughout are the compact matrix groups $T^n$, $U(n)$, $SU(n)$, $SO(n)$, and $Sp(n)$. These examples are not decorations; they are the testing ground for every general definition. The point of the theory is to explain why diagonal matrices, permutation symmetries, roots, weights, and characters organise the representation theory of compact groups. ## What Is the Structure Problem? What information is needed to recover the main features of a compact connected Lie group $G$ without handling all of $G$ at once? A compact group has enough averaging to make representation theory finite-dimensional and unitary in nature, while connectedness prevents the group from being assembled from several unrelated components. The structure problem asks how far we can reduce questions about $G$ to questions about a torus and a finite group of symmetries. The guiding picture is that a compact connected Lie group contains a maximal connected abelian subgroup $T$, called a maximal torus. For $U(n)$, this is the subgroup of diagonal unitary matrices; for $SU(n)$, it is the diagonal subgroup with determinant $1$. Conjugation by elements of $G$ moves tori around, and the Weyl group records the residual symmetries that preserve a chosen maximal torus. [definition: Compact Connected Lie Group] A compact connected Lie group is a Lie group $G$ whose underlying smooth manifold is compact and connected. [/definition] Compactness gives integration over the group, while connectedness makes the Lie algebra and the exponential map more controlling than they would be for a disconnected group. The course uses these hypotheses together: compactness supplies Haar measure and invariant inner products, and connectedness supports the conjugacy theorems for maximal tori. The first example is needed to make these hypotheses concrete inside the matrix groups used throughout the course. [example: Diagonal Torus In Special Unitary Group] In $SU(n)$, consider \begin{align*} T=\{\operatorname{diag}(z_1,\dots,z_n): |z_i|=1,\ z_1\cdots z_n=1\}. \end{align*} The map \begin{align*} (S^1)^{n-1}\to T,\qquad (z_1,\dots,z_{n-1})\mapsto \operatorname{diag}(z_1,\dots,z_{n-1},(z_1\cdots z_{n-1})^{-1}) \end{align*} is a [group isomorphism](/page/Group%20Isomorphism), since the determinant condition forces $z_n=(z_1\cdots z_{n-1})^{-1}$ and multiplication is componentwise. Hence $T$ is a torus of dimension $n-1$. Its Lie algebra is obtained by differentiating a path \begin{align*} \gamma(t)=\operatorname{diag}(e^{ia_1t},\dots,e^{ia_nt}) \end{align*} with $a_i\in \mathbb R$. The condition $\det\gamma(t)=1$ says \begin{align*} e^{i(a_1+\cdots+a_n)t}=1\text{ for all }t, \end{align*} so $a_1+\cdots+a_n=0$. Differentiating at $t=0$ gives \begin{align*} \gamma'(0)=\operatorname{diag}(ia_1,\dots,ia_n), \end{align*} which is diagonal, skew-Hermitian, and has trace $i(a_1+\cdots+a_n)=0$. Conversely, every diagonal skew-Hermitian trace-zero matrix has this form, so \begin{align*} \mathfrak t=\{\operatorname{diag}(ia_1,\dots,ia_n):a_i\in\mathbb R,\ a_1+\cdots+a_n=0\}. \end{align*} The torus is maximal: any connected abelian subgroup of $SU(n)$ containing $T$ consists of unitary matrices commuting with every diagonal matrix in $T$, and such matrices preserve each coordinate line, so they are diagonal; the determinant-one condition then puts them back in $T$. Permutation matrices conjugate diagonal matrices by permuting the diagonal entries, and after multiplying by a diagonal unitary matrix one may choose determinant $1$. Thus the Weyl group acts on $T$ by \begin{align*} \operatorname{diag}(z_1,\dots,z_n)\mapsto \operatorname{diag}(z_{\sigma^{-1}(1)},\dots,z_{\sigma^{-1}(n)}) \end{align*} for $\sigma\in S_n$, so the finite symmetry left after choosing the torus is permutation of the diagonal entries. [/example] This example already displays the main pattern. The continuous abelian part is the torus $T$, while the finite symmetry group is a permutation group. Chapters 4 and 5 turn this observation into roots and root data, and Chapter 9 uses it to compute characters. ## Why Compactness Changes Representation Theory Why should compact groups have a representation theory resembling finite group representation theory? The reason is averaging. Haar measure lets us average arbitrary choices over the group, producing invariant inner products and equivariant projections. [definition: Finite-Dimensional Unitary Representation] A finite-dimensional unitary representation of a Lie group $G$ is a homomorphism $\rho:G\to U(V)$, where $V$ is a finite-dimensional complex Hilbert space and $U(V)$ is the group of unitary linear maps $V\to V$. [/definition] This definition identifies the representation-theoretic setting in which orthogonal complements are available. The next theorem is needed because a general finite-dimensional representation of a compact group may not initially be presented as unitary, and the course needs a mechanism that makes unitarity available without changing the underlying representation. [quotetheorem:9712] [citeproof:9712] Once an invariant [inner product](/page/Inner%20Product) is available, invariant subspaces have invariant orthogonal complements. Thus the representation theory becomes semisimple in finite dimensions, just as in the finite group case. Compactness is essential here: for a noncompact group such as $(\mathbb R,+)$, the two-dimensional representation with $t$ acting by $e_1\mapsto e_1$ and $e_2\mapsto t e_1+e_2$ has no positive definite Hermitian inner product making every operator unitary, since non-identity unipotent operators cannot be unitary. The theorem also does not say that every representation is already unitary in its given coordinates; it says that the inner product can be changed without changing the underlying action. This is the analytic input behind complete reducibility, Schur orthogonality, and eventually Peter-Weyl theory. The next example is needed to connect the averaging proof with the familiar Fourier decomposition of a circle action. [example: Averaging On The Circle] Let $G=T^1$ act on a finite-dimensional complex [vector space](/page/Vector%20Space) $V$ by a continuous representation $\rho:T^1\to GL(V)$, and start with a Hermitian inner product $(\cdot,\cdot)_0$. Define \begin{align*} (v,w)_V=\frac{1}{2\pi}\int_0^{2\pi}(\rho(e^{i\theta})v,\rho(e^{i\theta})w)_0\,d\theta. \end{align*} The integral is Hermitian and linear in the same variable as $(\cdot,\cdot)_0$, because integration preserves addition and scalar multiplication. If $v\neq 0$, then the function $\theta\mapsto (\rho(e^{i\theta})v,\rho(e^{i\theta})v)_0$ is continuous and non-negative. Its value at $\theta=0$ is $(v,v)_0>0$, so it is positive on some interval around $0$, and therefore \begin{align*} (v,v)_V=\frac{1}{2\pi}\int_0^{2\pi}(\rho(e^{i\theta})v,\rho(e^{i\theta})v)_0\,d\theta>0. \end{align*} Thus $(\cdot,\cdot)_V$ is a Hermitian inner product. For invariance, fix $e^{i\phi}\in T^1$. Using that $\rho$ is a homomorphism, \begin{align*} (\rho(e^{i\phi})v,\rho(e^{i\phi})w)_V=\frac{1}{2\pi}\int_0^{2\pi}(\rho(e^{i(\theta+\phi)})v,\rho(e^{i(\theta+\phi)})w)_0\,d\theta. \end{align*} With $u=\theta+\phi$, this becomes \begin{align*} (\rho(e^{i\phi})v,\rho(e^{i\phi})w)_V=\frac{1}{2\pi}\int_\phi^{2\pi+\phi}(\rho(e^{iu})v,\rho(e^{iu})w)_0\,du. \end{align*} The integrand is $2\pi$-periodic because $e^{i(u+2\pi)}=e^{iu}$, so the last integral equals the integral over $[0,2\pi]$. Hence \begin{align*} (\rho(e^{i\phi})v,\rho(e^{i\phi})w)_V=(v,w)_V. \end{align*} Therefore every $\rho(e^{i\phi})$ is unitary for the averaged inner product. Since $T^1$ is abelian, these unitary operators commute with one another, and the finite-dimensional spectral theorem for commuting normal operators gives a simultaneous decomposition into common eigenspaces. The eigenvalue of $z\in T^1$ on such a common eigenspace is a continuous homomorphism $T^1\to S^1$, hence has the form $z\mapsto z^m$ for a unique $m\in\mathbb Z$. Thus the averaging construction makes the circle action unitary, and the representation splits into its Fourier weight spaces. [/example] The circle example is the abelian model for the whole course. General compact groups are not abelian, but their maximal tori are; the main challenge is to account for the extra Weyl group symmetry. ## Maximal Tori As a Coordinate System How can a non-abelian compact group be studied using an abelian subgroup? A maximal torus is not a coordinate chart in the manifold sense, since it has smaller dimension than $G$ in non-abelian cases. Instead, it is a conjugacy-theoretic coordinate system: every group element is conjugate into it, but the point of $T$ obtained is only determined up to Weyl group action. [definition: Torus] A torus is a compact connected abelian Lie group isomorphic to $(S^1)^r$ for some integer $r\geq 0$. [/definition] A torus has an elementary representation theory: its irreducible complex representations are one-dimensional characters. This makes tori computable, and the structural theorems explain why this computability is inherited, with corrections, by compact connected Lie groups. The next definition is needed to specify which torus inside $G$ carries as much abelian information as possible. [definition: Maximal Torus] A maximal torus of a Lie group $G$ is a torus $T\leq G$ that is not properly contained in any larger torus of $G$. [/definition] Maximality turns a torus from a convenient subgroup into a candidate for controlling conjugacy. The obstruction is that a small torus can miss most of the group, while two unrelated maximal choices would make any restriction-to-a-torus calculation depend on an arbitrary choice. The structural question is therefore whether compact connectedness forces every element to be represented in one maximal torus and whether all maximal tori give the same picture up to conjugation. [quotetheorem:9713] [citeproof:9713] This theorem justifies studying class functions on $G$ by restricting them to $T$, but its hypotheses are doing real work. If $G$ is disconnected, a component outside $G^\circ$ need not meet a chosen maximal torus in the identity component; finite extensions can add conjugacy classes invisible to that torus. If compactness is dropped, as in $SL(2,\mathbb R)$, not every element is conjugate into a compact torus: hyperbolic diagonal elements and elliptic rotations belong to different kinds of abelian subgroups. Even in the compact connected case the representative in $T$ is not unique, since points in the same Weyl group orbit are still conjugate in $G$. The next definition is needed to name the group that measures exactly this remaining ambiguity. [definition: Weyl Group] Let $T$ be a maximal torus of a compact connected Lie group $G$. The Weyl group of $(G,T)$ is \begin{align*} W(G,T)=N_G(T)/T, \end{align*} where $N_G(T)=\{g\in G:gTg^{-1}=T\}$ is the normaliser of $T$. [/definition] The quotient appears because elements of $T$ act on $T$ by conjugation as the identity. Thus the effective symmetry group acting on the torus is not the full normaliser but the finite quotient $N_G(T)/T$. The next example is needed because the Weyl group should be understood first as a concrete permutation group before it becomes an abstract reflection group. [example: Weyl Group Of Special Unitary Group] Let $T=\{\operatorname{diag}(z_1,\dots,z_n): |z_i|=1,\ z_1\cdots z_n=1\}\leq SU(n)$. If $A\in N_{SU(n)}(T)$, then $AtA^{-1}\in T$ for every $t\in T$. Choose $t=\operatorname{diag}(z_1,\dots,z_n)\in T$ with the $z_i$ pairwise distinct. The eigenspaces of $t$ are exactly the coordinate lines $\mathbb C e_1,\dots,\mathbb C e_n$, while the eigenspaces of $AtA^{-1}$ are $A\mathbb C e_1,\dots,A\mathbb C e_n$. Since $AtA^{-1}$ is diagonal, its eigenspaces are also the coordinate lines. Therefore there is a permutation $\sigma\in S_n$ such that \begin{align*} A\mathbb C e_j=\mathbb C e_{\sigma(j)} \end{align*} for each $j$. Thus $Ae_j=c_j e_{\sigma(j)}$ with $|c_j|=1$, because $A$ is unitary. Hence $A$ is a monomial unitary matrix: it has exactly one non-zero entry in each row and each column. Conversely, suppose $A$ is monomial unitary with determinant $1$, so $Ae_j=c_j e_{\sigma(j)}$ for some $\sigma\in S_n$ and unit complex numbers $c_j$. For $t=\operatorname{diag}(z_1,\dots,z_n)\in T$, we have \begin{align*} AtA^{-1}e_{\sigma(j)}=At(c_j^{-1}e_j)=A(c_j^{-1}z_j e_j)=z_j e_{\sigma(j)}. \end{align*} Thus $AtA^{-1}$ is diagonal, and its diagonal entry in position $k$ is $z_{\sigma^{-1}(k)}$. Since \begin{align*} z_{\sigma^{-1}(1)}\cdots z_{\sigma^{-1}(n)}=z_1\cdots z_n=1, \end{align*} we get $AtA^{-1}\in T$. Therefore $N_{SU(n)}(T)$ is exactly the group of determinant-one monomial unitary matrices. Define \begin{align*} \pi:N_{SU(n)}(T)\to S_n \end{align*} by sending a monomial matrix to its permutation of the coordinate lines. If $\pi(A)$ is the identity, then $Ae_j=c_j e_j$ for every $j$, so $A=\operatorname{diag}(c_1,\dots,c_n)$ and $\det A=c_1\cdots c_n=1$; hence $A\in T$. Thus $\ker\pi=T$. The map $\pi$ is surjective because for each $\sigma\in S_n$, the matrix $D_\sigma P_\sigma$ lies in $SU(n)$ when $P_\sigma$ is the permutation matrix and $D_\sigma=\operatorname{diag}(\det(P_\sigma)^{-1},1,\dots,1)$. Therefore \begin{align*} W(SU(n),T)=N_{SU(n)}(T)/T\cong S_n. \end{align*} Under this identification, \begin{align*} \operatorname{diag}(z_1,\dots,z_n)\mapsto \operatorname{diag}(z_{\sigma^{-1}(1)},\dots,z_{\sigma^{-1}(n)}). \end{align*} So the Weyl group remembers precisely the permutation ambiguity left after the diagonal torus has been chosen. [/example] This example explains why symmetric polynomials and alternating determinants enter the character theory of $SU(n)$. The Weyl group is the finite remnant of conjugation after the torus has been chosen. ## Roots, Weights, and Characters What replaces eigenvalues when the group is non-abelian but still controlled by a torus? The answer is that the torus acts by simultaneous diagonalisation on many vector spaces attached to $G$, and the resulting characters of $T$ are called weights. When the vector space is the complexified Lie algebra of $G$, the non-zero weights are the roots. [definition: Weight Of a Torus Representation] Let $T$ be a torus, let $V$ be a finite-dimensional complex vector space, and let $\rho:T\to GL(V)$ be a complex representation. A weight of $V$ with respect to $\rho$ is a continuous homomorphism $\lambda:T\to S^1$ such that \begin{align*} V_\lambda=\{v\in V:\rho(t)v=\lambda(t)v\text{ for all }t\in T\} \end{align*} is non-zero. [/definition] The weight space decomposition is the Fourier decomposition of a torus representation. For non-abelian $G$, weights of $T$ still appear after restricting a representation of $G$ to $T$. The next definition is needed to isolate the special weights that come from the adjoint action on the Lie algebra, since those weights control the geometry transverse to the maximal torus. [definition: Root] Let $G$ be a compact connected Lie group with maximal torus $T$ and Lie algebras $\mathfrak g$ and $\mathfrak t$. The adjoint representation restricts and complexifies to a representation $\operatorname{Ad}_{\mathbb C}|_T:T\to GL(\mathfrak g_{\mathbb C})$. A root is a non-zero weight for this representation. [/definition] Roots measure the failure of $G$ to be abelian. If $G=T$ is itself a torus, there are no roots; for $SU(n)$, roots record the ratios of diagonal entries acting on off-diagonal matrix units. The next example is needed to show that this abstract definition reproduces the familiar root system of type $A_{n-1}$. [example: Roots Of Special Unitary Group] For $G=SU(n)$, let \begin{align*} T=\{\operatorname{diag}(z_1,\dots,z_n): |z_i|=1,\ z_1\cdots z_n=1\}. \end{align*} The complexified Lie algebra is $\mathfrak{su}(n)_{\mathbb C}=\mathfrak{sl}_n(\mathbb C)$, so every $X=(x_{ab})\in \mathfrak{sl}_n(\mathbb C)$ splits as \begin{align*} X=\operatorname{diag}(x_{11},\dots,x_{nn})+\sum_{i\neq j}x_{ij}E_{ij}. \end{align*} Since $\operatorname{tr}X=0$, the diagonal part satisfies $x_{11}+\cdots+x_{nn}=0$. Thus \begin{align*} \mathfrak{sl}_n(\mathbb C)=\{\operatorname{diag}(a_1,\dots,a_n):a_1+\cdots+a_n=0\}\oplus \bigoplus_{i\neq j}\mathbb C E_{ij}. \end{align*} Now fix $t=\operatorname{diag}(z_1,\dots,z_n)\in T$. Since $t e_j=z_j e_j$ and $t^{-1}e_j=z_j^{-1}e_j$, for the matrix unit $E_{ij}$ we have \begin{align*} (tE_{ij}t^{-1})e_j=tE_{ij}(z_j^{-1}e_j)=t(z_j^{-1}e_i)=z_i z_j^{-1}e_i. \end{align*} If $k\neq j$, then $t^{-1}e_k=z_k^{-1}e_k$ and $E_{ij}e_k=0$, so \begin{align*} (tE_{ij}t^{-1})e_k=0. \end{align*} Therefore \begin{align*} \operatorname{Ad}_{\mathbb C}(t)(E_{ij})=tE_{ij}t^{-1}=z_i z_j^{-1}E_{ij}. \end{align*} On the diagonal trace-zero subspace, $tDt^{-1}=D$ because diagonal matrices commute, so the corresponding weight is the trivial character. The non-trivial weights are therefore \begin{align*} \alpha_{ij}(t)=z_i z_j^{-1}\qquad (i\neq j), \end{align*} with root space $\mathbb C E_{ij}$. Hence the roots of $SU(n)$ are exactly the characters $z\mapsto z_i z_j^{-1}$ for $i\neq j$, the type $A_{n-1}$ roots. [/example] The root system leads to a choice of positive roots, a Weyl chamber, and dominant weights. These choices are technical at first, but their purpose is concrete: they index irreducible representations and give formulas for their characters. To state what is eventually computed, we need the basic representation-theoretic invariant attached to any finite-dimensional representation. [definition: Character Of a Representation] Let $G$ be a compact Lie group and let $\rho:G\to GL(V)$ be a finite-dimensional complex representation. The character of $\rho$ is the function $\chi_\rho:G\to \mathbb C$ defined by \begin{align*} \chi_\rho(g)=\operatorname{tr}(\rho(g)). \end{align*} [/definition] [Characters are class functions](/theorems/5008), so the [maximal torus theorem](/theorems/9713) allows them to be studied on $T$. For a weight $\mu$, write $e^\mu$ for the corresponding character of $T$; after choosing positive roots, call a point of $T$ regular when it is not killed by any root factor in the Weyl denominator. The next theorem is needed because classification by highest weights becomes useful only after it gives an explicit formula for the character attached to each highest weight. [quotetheorem:9384] [citeproof:9384] The formula is the endpoint of the torus-and-Weyl-group philosophy, but it is not a formula for arbitrary weights. The weight $\lambda$ must be dominant and integral; otherwise there need not be an irreducible representation with highest weight $\lambda$, and the displayed alternating quotient may vanish or fail to be a character. The choice of positive roots is used to define dominance and $\rho$, although the final character is independent of this auxiliary choice once the highest weight convention is fixed. The quotient is first read on regular elements of $T$, where the denominator is non-zero, and then extended continuously across the singular set. Instead of computing traces on the full group, we compute finite sums in the weight lattice. ## Analytic Tools Used Later Which analytic inputs are needed for a course that is mostly structural? Three inputs recur: Haar measure, orthogonality of matrix coefficients, and integration over conjugacy classes. They allow representation theory, harmonic analysis, and geometry to communicate. [definition: Class Function] A class function on a group $G$ is a function $f:G\to \mathbb C$ satisfying \begin{align*} f(hgh^{-1})=f(g) \end{align*} for all $g,h\in G$. [/definition] Class functions are the functions naturally detected by characters. Since conjugacy classes meet a maximal torus, class functions are determined by their restriction to $T$, subject to Weyl group invariance. The next theorem is needed to turn this restriction principle into an integration formula with the correct root-theoretic density. [quotetheorem:9714] [citeproof:9714] This formula explains why the roots appear not only in representation theory but also in integration. The same denominator that occurs in the character formula is also the Jacobian of conjugation. Compactness is needed so that Haar measure has finite total mass and the change-of-variables argument gives an integration formula rather than a divergent volume comparison. Connectedness ensures that a single maximal torus meets every [conjugacy class](/page/Conjugacy%20Class); for disconnected compact groups, extra components require additional terms or separate analysis. The class-function hypothesis is also essential: a general function on $G$ is not determined by its values on $T$, so restriction to the torus cannot recover its integral. The factor $|W|^{-1}$ corrects the overcounting of regular conjugacy classes, whose intersections with $T$ are Weyl group orbits. The next theorem is needed because Weyl integration handles class functions, while harmonic analysis on $G$ requires a Fourier theory for all square-integrable functions. [quotetheorem:8833] [citeproof:8833] Peter-Weyl is the compact-group analogue of [Fourier series](/page/Fourier%20Series). When $G=T^n$, it recovers the usual Fourier basis indexed by the lattice $\mathbb Z^n$; for non-abelian $G$, the irreducible representations replace exponential functions. Compactness is again essential: for noncompact groups such as $\mathbb R$, the [Fourier transform](/page/Fourier%20Transform) involves a continuous spectrum rather than a discrete Hilbert [direct sum](/page/Direct%20Sum) of finite-dimensional representation spaces. The theorem also does not say that every $L^2$ function is a finite linear combination of matrix coefficients; it says finite sums are dense and arbitrary $L^2$ functions are obtained as Hilbert-space limits. This distinction is why later harmonic analysis separates algebraic computations with characters from analytic convergence questions. ## How the Course Fits Together How should the later chapters be read as a single argument rather than a list of topics? The course begins with compact averaging, then builds the conjugacy theory of maximal tori, then extracts the root and weight data that controls representations. The final chapters turn this data into integration and character formulas. The logical flow is as follows. Haar measure gives invariant inner products and complete reducibility. Maximal tori reduce conjugacy questions to abelian subgroups. The Weyl group records the finite ambiguity left by this reduction. Roots describe the infinitesimal action transverse to the torus. Weights classify how representations decompose on the torus. Characters package representations into computable class functions. [remark: Running Philosophy] A compact connected Lie group should be treated as a torus together with extra finite and root-theoretic data. The torus supplies the lattice, the Weyl group supplies symmetry, and the roots supply the geometry transverse to the torus. [/remark] This philosophy is powerful because it is computational. For $SU(2)$, the maximal torus is a circle, the Weyl group has order $2$, the roots consist of two opposite weights, and irreducible representations are indexed by non-negative integers. For $SU(n)$, the same ideas produce partitions, Schur polynomials, and the familiar representation theory of the special unitary groups. The final example is needed to make the whole course outline visible in the smallest non-abelian compact group. The same structures also connect the course to other areas. Topologically, maximal tori and Weyl groups control much of the cohomology and characteristic-class theory of compact Lie groups and their homogeneous spaces. Algebraically, the character formulas for $U(n)$ and $SU(n)$ recover symmetric functions such as Schur polynomials. In physics, compact groups such as $SU(2)$ and $SU(3)$ organise angular momentum and internal symmetries through their finite-dimensional representations. [example: The Course In The Case Of SU(2)] For $G=SU(2)$, take \begin{align*} T=\{\operatorname{diag}(z,z^{-1}):|z|=1\}. \end{align*} Let $w$ be the [linear map](/page/Linear%20Map) with $w e_1=e_2$ and $w e_2=-e_1$. Its columns are orthonormal and its determinant is $1$, so $w\in SU(2)$. For $t=\operatorname{diag}(z,z^{-1})$, the inverse satisfies $w^{-1}e_1=-e_2$ and $w^{-1}e_2=e_1$. Hence \begin{align*} wtw^{-1}e_1=wt(-e_2)=w(-z^{-1}e_2)=z^{-1}e_1. \end{align*} Also \begin{align*} wtw^{-1}e_2=wt(e_1)=w(ze_1)=ze_2. \end{align*} Therefore \begin{align*} wtw^{-1}=\operatorname{diag}(z^{-1},z), \end{align*} so the non-trivial Weyl group element acts on $T$ by $z\mapsto z^{-1}$. To compute the roots, use the standard basis $H=E_{11}-E_{22}$, $E=E_{12}$, and $F=E_{21}$ of $\mathfrak{sl}_2(\mathbb C)$. Since diagonal matrices commute, $tHt^{-1}=H$. For $E=E_{12}$, we have $E e_2=e_1$ and $E e_1=0$, so \begin{align*} tEt^{-1}e_2=tE(ze_2)=zt e_1=z^2e_1. \end{align*} Also $tEt^{-1}e_1=tE(z^{-1}e_1)=0$, hence \begin{align*} tEt^{-1}=z^2E. \end{align*} Similarly, $F e_1=e_2$ and $F e_2=0$, so \begin{align*} tFt^{-1}e_1=tF(z^{-1}e_1)=z^{-1}t e_2=z^{-2}e_2. \end{align*} Also $tFt^{-1}e_2=tF(ze_2)=0$, hence \begin{align*} tFt^{-1}=z^{-2}F. \end{align*} Thus the two non-zero weights of the adjoint action are the roots $z\mapsto z^2$ and $z\mapsto z^{-2}$. With the usual highest-weight indexing, the irreducible representation of highest weight $m\geq 0$ is $\operatorname{Sym}^m(\mathbb C^2)$. If $e_1,e_2$ are the standard basis vectors, then $t e_1=ze_1$ and $t e_2=z^{-1}e_2$. A basis of $\operatorname{Sym}^m(\mathbb C^2)$ is given by the monomials \begin{align*} e_1^{m-k}e_2^k\qquad 0\leq k\leq m. \end{align*} On this basis vector, \begin{align*} t(e_1^{m-k}e_2^k)=(ze_1)^{m-k}(z^{-1}e_2)^k=z^{m-k}z^{-k}e_1^{m-k}e_2^k=z^{m-2k}e_1^{m-k}e_2^k. \end{align*} The trace is the sum of these diagonal eigenvalues, so \begin{align*} \chi_m(z)=\sum_{k=0}^m z^{m-2k}=z^m+z^{m-2}+\cdots+z^{-m}. \end{align*} Thus in $SU(2)$ the maximal torus is a circle, the Weyl group has one non-trivial inversion symmetry, the roots are the two opposite weights $\pm 2$, and the characters are finite symmetric strings of torus weights. [/example] The introduction has therefore set the agenda: compactness gives averaging, maximal tori give abelian structure, Weyl groups give finite symmetry, roots and weights give linear algebraic data, and characters turn the classification into explicit functions. Chapter 1 now begins the detailed development of the course: compactness, maximal tori, Weyl symmetry, roots and weights, and characters. The next chapter shifts from this structural overview to the analytic machinery that compactness makes available, beginning with Haar measure and averaging. # 1. Haar Analysis on Compact Lie Groups This opening chapter sets up the analytic tools that will be used throughout the structure theory of compact Lie groups. Compactness lets us average over the group, and that single operation turns arbitrary finite-dimensional representations into unitary ones, projections into equivariant projections, and functions into class functions. The later theory of maximal tori, Weyl groups, and characters depends on this analytic foundation: characters are first honest functions on the group, and their orthogonality is a consequence of Haar integration. ## Averaging over a Compact Group How can a compact Lie group be treated like a finite group when it has infinitely many elements? For a finite group $G$, the basic averaging operator is \begin{align*} \frac{1}{|G|}\sum_{g \in G}. \end{align*} For a compact Lie group, the replacement is integration against a measure whose invariance expresses the idea that every group element is counted with the same weight. [definition: Left Haar Measure] Let $G$ be a compact Lie group, and let $\mathcal B(G)$ be its Borel $\sigma$-algebra. A left Haar measure on $G$ is a nonzero regular Borel measure \begin{align*} \mu:\mathcal B(G)\to [0,\infty] \end{align*} such that \begin{align*} \mu(gA)=\mu(A) \end{align*} for every Borel set $A \subset G$ and every $g \in G$. [/definition] The condition says that translating a measurable set on the left does not change its measure, so integration can respect the group law rather than an external coordinate choice. For averaging arguments we also need a fixed scale, since multiplying a Haar measure by a positive constant preserves left-invariance. On a compact group there is a canonical normalization, obtained by scaling so that the whole group has total mass $1$. [definition: Normalized Haar Measure] Let $G$ be a compact Lie group, and let $\mathcal B(G)$ be its Borel $\sigma$-algebra. The normalized Haar measure on $G$ is a left Haar measure \begin{align*} dg:\mathcal B(G)\to [0,\infty] \end{align*} satisfying \begin{align*} \int_G 1\,dg=1. \end{align*} [/definition] This normalization is the analogue of dividing by $|G|$ for a finite group, and it is what makes group averages genuine averages rather than arbitrary multiples. The notation $dg$ will be used throughout the course, but this notation is useful only if such a measure exists and is independent of hidden choices. The next theorem supplies exactly that foundation and records the invariance properties used in calculations. [quotetheorem:1063] [citeproof:1063] The compactness hypothesis is doing real work here: on $\mathbb R$ there is translation-invariant [Lebesgue measure](/page/Lebesgue%20Measure), but no nonzero translation-invariant probability measure, so the normalized averaging operator has no analogue. The theorem also does not say that Haar integration is a coordinate formula; even for matrix groups, the measure is characterized by invariance rather than by a preferred parametrization. What it gives is exactly the operation needed later: any construction formed by integrating all group translates has a chance to become independent of the original choice. [example: Averaging a Function on the Circle] Let $G=S^1$, write $z=e^{i\theta}$ with $0\leq \theta<2\pi$, and let $f:S^1\to\mathbb C$ be continuous. Under this parametrization the normalized Haar integral is \begin{align*} \int_{S^1} f(z)\,dz=\frac{1}{2\pi}\int_0^{2\pi} f(e^{i\theta})\,d\theta. \end{align*} To see the rotation-invariance in coordinates, fix $w=e^{i\phi}\in S^1$ and set $F(t)=f(e^{it})$. Then $F(t+2\pi)=F(t)$, and \begin{align*} \frac{1}{2\pi}\int_0^{2\pi} f(we^{i\theta})\,d\theta=\frac{1}{2\pi}\int_0^{2\pi} f(e^{i(\phi+\theta)})\,d\theta. \end{align*} With the change of variables $t=\phi+\theta$, this becomes \begin{align*} \frac{1}{2\pi}\int_{\phi}^{\phi+2\pi} F(t)\,dt. \end{align*} Using $2\pi$-periodicity, \begin{align*} \int_{\phi}^{\phi+2\pi}F(t)\,dt=\int_{\phi}^{2\pi}F(t)\,dt+\int_{2\pi}^{\phi+2\pi}F(t)\,dt. \end{align*} In the second integral, put $u=t-2\pi$; then $F(t)=F(u)$ and \begin{align*} \int_{2\pi}^{\phi+2\pi}F(t)\,dt=\int_0^\phi F(u)\,du. \end{align*} Therefore \begin{align*} \int_{\phi}^{\phi+2\pi}F(t)\,dt=\int_0^{2\pi}F(t)\,dt. \end{align*} Hence \begin{align*} \int_{S^1} f(wz)\,dz=\int_{S^1} f(z)\,dz. \end{align*} Thus Haar averaging on $S^1$ is exactly averaging over one full period, and rotating the variable only changes where that period starts. [/example] For non-abelian compact groups, Haar measure gives the same averaging principle but with multiplication replacing addition. The next example is the prototype used later when studying $SU(n)$-representations. [example: Averaging over SU(n)] Let $n\geq 2$, let $G=SU(n)$, and let $F:SU(n)\to\mathbb C$ be continuous. By left- and right-invariance of normalized Haar measure, as in *Existence and Uniqueness of Normalized Haar Measure*, for every fixed $h\in SU(n)$ we have \begin{align*} \int_{SU(n)}F(hg)\,dg=\int_{SU(n)}F(g)\,dg. \end{align*} The same theorem gives \begin{align*} \int_{SU(n)}F(gh)\,dg=\int_{SU(n)}F(g)\,dg. \end{align*} Thus the average \begin{align*} A(F)=\int_{SU(n)}F(g)\,dg \end{align*} is unchanged by left or right translation of the argument. Now fix a complex $n\times n$ matrix $A$ and put \begin{align*} I_A=\int_{SU(n)}\operatorname{tr}(Ag)\,dg. \end{align*} Choose $\zeta\in S^1$ such that $\zeta^n=1$ and $\zeta\neq 1$, for example $\zeta=e^{2\pi i/n}$. Since $\det(\zeta I_n)=\zeta^n\det(I_n)=1$, the scalar matrix $\zeta I_n$ lies in $SU(n)$. Applying left-invariance to the [continuous function](/page/Continuous%20Function) $g\mapsto \operatorname{tr}(Ag)$ gives \begin{align*} I_A=\int_{SU(n)}\operatorname{tr}(A(\zeta I_n)g)\,dg. \end{align*} Because $(\zeta I_n)g=\zeta g$, we have $A(\zeta I_n)g=A(\zeta g)=\zeta Ag$. By linearity of trace, \begin{align*} \operatorname{tr}(A(\zeta I_n)g)=\operatorname{tr}(\zeta Ag)=\zeta\operatorname{tr}(Ag). \end{align*} Therefore \begin{align*} I_A=\int_{SU(n)}\zeta\operatorname{tr}(Ag)\,dg. \end{align*} Since $\zeta$ is constant in $g$, linearity of the integral gives \begin{align*} I_A=\zeta\int_{SU(n)}\operatorname{tr}(Ag)\,dg=\zeta I_A. \end{align*} Hence \begin{align*} (1-\zeta)I_A=0. \end{align*} Since $\zeta\neq 1$, it follows that \begin{align*} I_A=0. \end{align*} So this linear matrix coefficient has zero Haar average: the averaging process keeps only the part of a function that is invariant under all translations, foreshadowing Schur orthogonality. [/example] ## Invariant Inner Products and Unitary Representations Why does compactness make representation theory behave like finite-dimensional linear algebra with orthogonal decompositions? The answer is that an arbitrary Hermitian inner product can be averaged over the group, producing an equivalent inner product for which the group acts by unitary maps. [definition: Finite-Dimensional Representation] Let $G$ be a Lie group and let $V$ be a finite-dimensional complex vector space. A finite-dimensional complex representation of $G$ on $V$ is a smooth [group homomorphism](/page/Group%20Homomorphism) \begin{align*} \rho:G\to GL(V). \end{align*} [/definition] This definition turns group elements into linear transformations, but it does not yet impose any compatibility with an inner product. Orthogonality is the mechanism that lets invariant subspaces split cleanly. The next definition names the representations for which every group element preserves the inner product. [definition: Unitary Representation] Let $G$ be a Lie group and let $V$ be a finite-dimensional complex [inner product space](/page/Inner%20Product%20Space) with inner product $(\cdot,\cdot)_V$, linear in the first argument. A representation $\rho:G\to GL(V)$ is unitary if \begin{align*} (\rho(g)v,\rho(g)w)_V=(v,w)_V \end{align*} for all $g\in G$ and all $v,w\in V$. [/definition] Unitary representations are valuable because invariant subspaces automatically have invariant orthogonal complements. At first this may seem like an additional hypothesis on the representation, depending on a fortunate choice of inner product. This raises the key compact-group problem: given an arbitrary representation, can Haar measure build the missing invariant inner product? [quotetheorem:9712] [citeproof:9712] Compactness is essential: for the noncompact group $\mathbb R$ acting on $\mathbb C^2$ by sending $e_1$ to $e_1$ and $e_2$ to $t e_1+e_2$, no positive definite Hermitian inner product can make all operators unitary, since a nontrivial unipotent operator is not unitary for any positive definite form. The theorem also does not say that the starting inner product was already invariant, or that the averaged inner product is unique when a representation has several irreducible pieces. Its importance is that representation-theoretic questions can now be translated into unitary linear algebra. To describe the pieces that appear in an [orthogonal decomposition](/theorems/436), we need the representation-theoretic analogue of a vector space with no smaller invariant part. This leads to irreducibility. [definition: Irreducible Representation] Let $G$ be a Lie group and let $\rho:G\to GL(V)$ be a finite-dimensional complex representation. The representation is irreducible if $V\neq 0$ and the only $G$-invariant complex subspaces of $V$ are $0$ and $V$. [/definition] The definition isolates representations that cannot be decomposed further by invariant subspaces. For a general group, the existence of an invariant subspace does not guarantee a complementary invariant subspace, so reducible representations may still fail to split. Compactness eliminates this obstruction through the invariant inner product just constructed. [quotetheorem:8826] [citeproof:8826] The contrast with noncompact groups is important: for the representation of $\mathbb R$ on $\mathbb C^2$ sending $e_1$ to $e_1$ and $e_2$ to $t e_1+e_2$, the line spanned by $e_1$ is invariant, but it has no invariant complementary line. Thus reducibility need not imply decomposability without compactness. The theorem also does not give a canonical decomposition; irreducible summands may occur with multiplicity, and choices remain inside each isotypic component. What compactness supplies is the missing orthogonal complement, and this is the structural input behind [Schur's lemma](/theorems/2414) and the orthogonality relations that follow. [example: Averaging an Inner Product over SU(n)] Let $\rho:SU(n)\to GL(V)$ be a finite-dimensional complex representation, and start with a Hermitian inner product $(\cdot,\cdot)_0$ on $V$. Define \begin{align*} (v,w)_{SU(n)}=\int_{SU(n)}(\rho(g)v,\rho(g)w)_0\,dg. \end{align*} For $h\in SU(n)$, the representation identity gives $\rho(g)\rho(h)=\rho(gh)$, so \begin{align*} (\rho(h)v,\rho(h)w)_{SU(n)}=\int_{SU(n)}(\rho(gh)v,\rho(gh)w)_0\,dg. \end{align*} By right-invariance of normalized Haar measure, as in *[Averaging Inner Products](/theorems/9712)*, the substitution $u=gh$ does not change the integral. Hence \begin{align*} (\rho(h)v,\rho(h)w)_{SU(n)}=\int_{SU(n)}(\rho(u)v,\rho(u)w)_0\,du=(v,w)_{SU(n)}. \end{align*} Thus the averaged product is $SU(n)$-invariant. For the defining representation on $V=\mathbb C^n$, take the usual Hermitian product \begin{align*} (v,w)_0=\sum_{j=1}^n v_j\overline{w_j}. \end{align*} If $U=(U_{ij})\in SU(n)$, then $U^*U=I$, so $\sum_{i=1}^n U_{ij}\overline{U_{ik}}=\delta_{jk}$. Therefore \begin{align*} (Uv,Uw)_0=\sum_{i=1}^n\left(\sum_{j=1}^nU_{ij}v_j\right)\overline{\left(\sum_{k=1}^nU_{ik}w_k\right)}. \end{align*} Expanding the conjugate and regrouping the finite sums gives \begin{align*} (Uv,Uw)_0=\sum_{j=1}^n\sum_{k=1}^n v_j\overline{w_k}\left(\sum_{i=1}^nU_{ij}\overline{U_{ik}}\right). \end{align*} Using $\sum_iU_{ij}\overline{U_{ik}}=\delta_{jk}$, \begin{align*} (Uv,Uw)_0=\sum_{j=1}^n\sum_{k=1}^n v_j\overline{w_k}\delta_{jk}=\sum_{j=1}^n v_j\overline{w_j}=(v,w)_0. \end{align*} Thus the integrand $(Uv,Uw)_0$ is the constant function $(v,w)_0$, and normalization gives \begin{align*} (v,w)_{SU(n)}=\int_{SU(n)}(v,w)_0\,dU=(v,w)_0\int_{SU(n)}1\,dU=(v,w)_0. \end{align*} So averaging returns the usual Hermitian product in the defining representation. For a tensor representation such as $V=\operatorname{Sym}^k(\mathbb C^n)$, the same formula averages any starting Hermitian product into an invariant one. With that product, invariant subspaces have invariant orthogonal complements, which is the linear-algebra mechanism used to decompose tensor powers. [/example] ## Matrix Coefficients and Schur Orthogonality Once representations are unitary, their entries behave like Fourier modes. The analogue of orthogonality of exponentials on the circle is Schur orthogonality for matrix coefficients of irreducible compact-group representations. [definition: Matrix Coefficient] Let $G$ be a compact Lie group, let $(\rho,V)$ be a finite-dimensional complex representation, and let $\lambda\in V^*$. For $v\in V$, the associated matrix coefficient is the function $c_{\lambda,v}:G\to\mathbb C$ given by \begin{align*} c_{\lambda,v}(g)=\lambda(\rho(g)v). \end{align*} [/definition] For unitary representations, choosing an [orthonormal basis](/page/Orthonormal%20Basis) converts this definition into ordinary matrix entries. These functions are the basic waves of compact-group Fourier analysis, so the next question is how their $L^2(G)$ inner products behave. The answer is the matrix-coefficient form of Schur orthogonality. [quotetheorem:9715] [citeproof:9715] Schur orthogonality is the first form of harmonic analysis on compact groups, but each hypothesis has content. Irreducibility is needed because if $\rho=\chi_1\oplus \chi_2$ on $S^1$, its diagonal coefficients include two different Fourier modes and the coefficient block is not governed by a single factor $1/\dim V$. Unitarity is needed for the displayed formula with complex conjugates: after changing to a non-orthonormal basis, the same representation has matrix entries whose $L^2$ inner products contain the Gram matrix of the basis rather than Kronecker deltas. Compactness supplies the normalized Haar integral; for $\mathbb R$, the characters $t\mapsto e^{i\lambda t}$ are not square-integrable against Lebesgue measure and their orthogonality is distributional rather than an equality of finite integrals. What the theorem provides is the non-abelian analogue of orthogonal Fourier modes: irreducible representations supply mutually orthogonal blocks of functions on $G$. This is the finite-dimensional shadow of the Peter-Weyl theorem, where these matrix coefficients become dense in $L^2(G)$. [example: Fourier Series on the Torus] Let $G=\mathbb T^n=(S^1)^n$. Its irreducible unitary representations are the characters \begin{align*} \chi_m(z_1,\dots,z_n)=z_1^{m_1}\cdots z_n^{m_n}, \qquad m=(m_1,\dots,m_n)\in\mathbb Z^n. \end{align*} We compute the Haar inner product of two such characters. Write $z_j=e^{i\theta_j}$ with $0\leq \theta_j<2\pi$. Since $\overline{z_j^{m'_j}}=z_j^{-m'_j}$ on $S^1$, we have \begin{align*} \chi_m(z)\overline{\chi_{m'}(z)}=\prod_{j=1}^n z_j^{m_j-m'_j}. \end{align*} Under the product normalized Haar measure on $\mathbb T^n$, \begin{align*} \int_{\mathbb T^n}\chi_m(z)\overline{\chi_{m'}(z)}\,dz=\frac{1}{(2\pi)^n}\int_0^{2\pi}\cdots\int_0^{2\pi}\prod_{j=1}^n e^{i(m_j-m'_j)\theta_j}\,d\theta_1\cdots d\theta_n. \end{align*} Because the integrand is a product of one-variable functions, [Fubini's theorem](/theorems/2961) gives \begin{align*} \int_{\mathbb T^n}\chi_m(z)\overline{\chi_{m'}(z)}\,dz=\prod_{j=1}^n\left(\frac{1}{2\pi}\int_0^{2\pi} e^{i(m_j-m'_j)\theta}\,d\theta\right). \end{align*} For an integer $\ell$, the one-dimensional factor is $1$ if $\ell=0$. If $\ell\neq 0$, then \begin{align*} \frac{1}{2\pi}\int_0^{2\pi} e^{i\ell\theta}\,d\theta=\frac{1}{2\pi}\frac{e^{2\pi i\ell}-1}{i\ell}=0, \end{align*} since $e^{2\pi i\ell}=1$. Therefore each factor equals $\delta_{m_j,m'_j}$, so \begin{align*} \int_{\mathbb T^n}\chi_m(z)\overline{\chi_{m'}(z)}\,dz=\prod_{j=1}^n\delta_{m_j,m'_j}=\delta_{m,m'}. \end{align*} Thus ordinary Fourier series on $\mathbb T^n$ are the abelian model for the general compact-group theory: the characters are mutually orthogonal Fourier modes. [/example] The torus example explains why compact representation theory should be read as non-abelian Fourier analysis. In the non-abelian case, single characters are replaced by matrix coefficients, and commutativity is replaced by the bookkeeping of irreducible representations. ## Class Functions and Convolution Which functions on a compact group are visible from conjugacy classes rather than individual elements? This question leads to class functions, convolution, and central functions, all of which will reappear when maximal tori and Weyl groups enter the course. [definition: Class Function] Let $G$ be a group. A function $f:G\to\mathbb C$ is a class function if \begin{align*} f(hgh^{-1})=f(g) \end{align*} for all $g,h\in G$. [/definition] Class functions are the functions that forget the position of an element inside its conjugacy class. A representation assigns a linear operator to each group element, but the operator itself depends on a chosen basis and contains more information than a conjugacy-invariant function should remember. The invariant numerical datum we need is obtained by taking the trace: it removes the basis choice, is stable under conjugation, and still records enough of the representation to support Fourier-style decomposition later. [definition: Character of a Representation] Let $G$ be a compact Lie group and let $\rho:G\to GL(V)$ be a finite-dimensional complex representation. The character of $\rho$ is the function $\chi_\rho:G\to\mathbb C$ defined by \begin{align*} \chi_\rho(g)=\operatorname{tr}(\rho(g)). \end{align*} [/definition] The trace identity $\operatorname{tr}(ABA^{-1})=\operatorname{tr}(B)$ shows that characters are class functions. Later chapters will show that for compact connected Lie groups, much of a character is determined by its restriction to a maximal torus. [example: Class Functions on SU(2)] Let $A\in SU(2)$. Since $A$ is unitary, the finite-dimensional spectral theorem gives an orthonormal basis in which $A$ is diagonal with eigenvalues $\lambda_1,\lambda_2\in S^1$. Because $A\in SU(2)$, its determinant is $1$, so \begin{align*} \lambda_1\lambda_2=1. \end{align*} Choose $\theta\in\mathbb R$ with $\lambda_1=e^{i\theta}$. Then \begin{align*} \lambda_2=\lambda_1^{-1}=e^{-i\theta}. \end{align*} After adjusting the diagonalizing unitary matrix by a scalar of modulus $1$, the change-of-basis matrix may be chosen in $SU(2)$, so $A$ is conjugate inside $SU(2)$ to $\operatorname{diag}(e^{i\theta},e^{-i\theta})$. The parameter $\theta$ is not unique. Replacing $\theta$ by $\theta+2\pi k$ does not change the diagonal matrix, because \begin{align*} e^{i(\theta+2\pi k)}=e^{i\theta}e^{2\pi i k}=e^{i\theta} \end{align*} for $k\in\mathbb Z$, and similarly $e^{-i(\theta+2\pi k)}=e^{-i\theta}$. Also, the matrices $\operatorname{diag}(e^{i\theta},e^{-i\theta})$ and $\operatorname{diag}(e^{-i\theta},e^{i\theta})$ are conjugate in $SU(2)$: the linear map $s$ defined by $s(e_1)=e_2$ and $s(e_2)=-e_1$ has determinant $1$, and it swaps the two eigenspaces. Thus a class function on $SU(2)$ is determined by a function of $\theta$ that is $2\pi$-periodic and unchanged under $\theta\mapsto -\theta$. For the defining representation on $\mathbb C^2$, the character at this diagonal representative is the trace: \begin{align*} \chi(\theta)=\operatorname{tr}(\operatorname{diag}(e^{i\theta},e^{-i\theta})). \end{align*} The trace of a diagonal matrix is the sum of its diagonal entries, so \begin{align*} \chi(\theta)=e^{i\theta}+e^{-i\theta}. \end{align*} Using Euler's formulas $e^{i\theta}=\cos\theta+i\sin\theta$ and $e^{-i\theta}=\cos\theta-i\sin\theta$, we get \begin{align*} e^{i\theta}+e^{-i\theta}=(\cos\theta+i\sin\theta)+(\cos\theta-i\sin\theta)=2\cos\theta. \end{align*} Thus $\chi(\theta)=2\cos\theta$, an even $2\pi$-periodic function. This is the first appearance of the Weyl symmetry for $SU(2)$: conjugacy identifies $\theta$ with $-\theta$. [/example] The example shows that conjugacy collapses a non-[abelian group](/page/Abelian%20Group) to a smaller parameter space, but we also need an operation on functions that remembers group multiplication. Convolution packages multiplication in the group into an operation on functions. On an abelian compact group it is the usual convolution from Fourier analysis; on a non-abelian compact group the order of multiplication matters. [definition: Convolution on a Compact Group] Let $G$ be a compact Lie group and let $f_1,f_2\in C(G)$. Their convolution is the function $f_1*f_2:G\to\mathbb C$ defined by \begin{align*} (f_1*f_2)(g)=\int_G f_1(x)f_2(x^{-1}g)\,dx. \end{align*} [/definition] Convolution is the function-level shadow of composition of operators, and it interacts especially well with functions that are constant on conjugacy classes. This motivates naming the class functions that play the role of central elements for convolution. The terminology emphasizes their algebraic role rather than introducing a new condition. [definition: Central Function] Let $G$ be a compact Lie group. A continuous function $f:G\to\mathbb C$ is central if it is a class function. [/definition] The word central is used because these functions form the centre of the convolution [algebra of continuous functions](/theorems/197), after passing to the appropriate dense algebraic setting or to $L^2$ convolution where the products are defined. Since characters are the central functions attached to irreducible representations, it is natural to ask whether the orthogonality of matrix coefficients descends to characters. The next theorem gives the character-level orthogonality relation. [quotetheorem:9716] [citeproof:9716] This result is the character-level form of Schur orthogonality, and its hypotheses are still restrictive. Irreducibility is essential: if $\rho=\tau\oplus\tau$, then $\chi_\rho=2\chi_\tau$ and $\int_G|\chi_\rho(g)|^2\,dg=4$, so the norm records multiplicity rather than equalling $1$. Unitarity is not a separate loss of generality for compact groups because every finite-dimensional representation can be unitarized by averaging, but the proof uses unitary matrix coefficients to apply Schur orthogonality in the displayed form. Compactness is also essential: on a noncompact group such as $\mathbb R$, the characters $t\mapsto e^{i\lambda t}$ are not elements of $L^2(\mathbb R)$ with respect to Lebesgue measure, so there is no normalized character inner product of the same kind. The theorem gives the orthonormality part of the story; it does not say that every class function is already a finite linear combination of irreducible characters. That completeness statement belongs to the Peter-Weyl theory and requires a density or Hilbert-space closure assertion. Later this orthonormality will combine with the [Weyl integration formula](/theorems/9714) to turn integrals over $G$ into integrals over a maximal torus. [example: Convolution of Characters on the Circle] Let $G=S^1$ and let $\chi_m(z)=z^m$ for $m\in\mathbb Z$. Since $S^1$ is abelian, $hzh^{-1}=z$ for all $h,z\in S^1$, so every continuous function on $S^1$ is a class function. For $z\in S^1$, the definition of convolution gives \begin{align*} (\chi_m*\chi_{m'})(z)=\int_{S^1}\chi_m(x)\chi_{m'}(x^{-1}z)\,dx. \end{align*} Substituting $\chi_m(x)=x^m$ and $\chi_{m'}(x^{-1}z)=(x^{-1}z)^{m'}$ gives \begin{align*} (\chi_m*\chi_{m'})(z)=\int_{S^1}x^m(x^{-1}z)^{m'}\,dx. \end{align*} Because $S^1$ is abelian, powers multiply in the usual way: \begin{align*} x^m(x^{-1}z)^{m'}=x^m x^{-m'}z^{m'}=x^{m-m'}z^{m'}. \end{align*} Since $z^{m'}$ is constant with respect to the integration variable $x$, linearity of the integral gives \begin{align*} (\chi_m*\chi_{m'})(z)=z^{m'}\int_{S^1}x^{m-m'}\,dx. \end{align*} It remains to evaluate the last integral. Write $x=e^{i\theta}$ with $0\leq\theta<2\pi$. For $\ell=m-m'$, normalized Haar measure on $S^1$ gives \begin{align*} \int_{S^1}x^\ell\,dx=\frac{1}{2\pi}\int_0^{2\pi}e^{i\ell\theta}\,d\theta. \end{align*} If $\ell=0$, then $e^{i\ell\theta}=1$, so \begin{align*} \frac{1}{2\pi}\int_0^{2\pi}e^{i\ell\theta}\,d\theta=\frac{1}{2\pi}\int_0^{2\pi}1\,d\theta=1. \end{align*} If $\ell\neq 0$, then an antiderivative of $e^{i\ell\theta}$ is $e^{i\ell\theta}/(i\ell)$, hence \begin{align*} \frac{1}{2\pi}\int_0^{2\pi}e^{i\ell\theta}\,d\theta=\frac{1}{2\pi}\left(\frac{e^{2\pi i\ell}-1}{i\ell}\right)=0, \end{align*} because $\ell\in\mathbb Z$ implies $e^{2\pi i\ell}=1$. Therefore \begin{align*} \int_{S^1}x^{m-m'}\,dx=\delta_{m,m'}. \end{align*} Substituting this into the convolution formula gives \begin{align*} (\chi_m*\chi_{m'})(z)=z^{m'}\delta_{m,m'}. \end{align*} Equivalently, $\chi_m*\chi_{m'}=\chi_m$ when $m=m'$ and $\chi_m*\chi_{m'}=0$ when $m\neq m'$. Thus convolution acts diagonally on Fourier modes, the abelian version of the projection behaviour encoded by matrix coefficients for general compact groups. [/example] The chapter has established the analytic language needed for compact Lie groups: normalized Haar measure, invariant inner products, irreducible unitary representations, matrix coefficient orthogonality, and class functions. The next chapter begins the structural part of the course by showing that in a compact connected Lie group, maximal tori play the role of the diagonal subgroup in $SU(n)$, and conjugacy reduces many questions to those tori. Haar analysis gives the representation-theoretic tools that make compact groups tractable: invariant integration, unitary structures, and orthogonality of matrix coefficients. With that analytic foundation in place, we can move from general averaging principles to the geometry of maximal tori and the conjugacy theorems they control. # 2. Maximal Tori and the Conjugacy Theorems This chapter begins the structural study of compact connected Lie groups by isolating the abelian pieces that control the group. In Chapter 1, Haar averaging showed that compactness makes representation theory and invariant geometry well behaved. Here compactness is used in a different way: it forces large connected abelian subgroups to be tori, and it makes them large enough that every element of the group is conjugate into one of them. The main objects are maximal tori. A maximal torus is not merely a convenient abelian subgroup; it is the place where the Weyl group of Chapter 3, the roots of Chapter 4, the weights of Chapters 5 and 6, and the integration and character formulae of Chapters 8 and 9 will live. The conjugacy theorems say that the choice of maximal torus is essentially unique up to conjugation, so constructions made on one maximal torus can be transported across the group. ## Tori in Compact Matrix Groups and Their Lie Algebras The first question is what a connected abelian compact Lie group can look like. For matrix groups the answer is visible from linear algebra: commuting unitary matrices can be diagonalised simultaneously, and their eigenvalues lie on the unit circle. This section records the model case and fixes the Lie algebra conventions used throughout the course. A torus is the compact Lie group analogue of a finite-dimensional real vector space modulo a lattice. [definition: Torus] A torus is a Lie group isomorphic to $(S^1)^r$ for some integer $r \ge 0$. The integer $r$ is called its dimension or rank. [/definition] The case $r=0$ is the one-point group. For $r>0$, the Lie algebra of $(S^1)^r$ is naturally $i\mathbb R^r$ when $S^1\subset \mathbb C^\times$, and the exponential map is taken componentwise. The next example makes the quotient-by-a-lattice picture explicit, which is the local model behind every torus used later. [example: Standard Torus] Let $T^r=(S^1)^r\subset(\mathbb C^\times)^r$, so its Lie algebra is $\mathfrak t=i\mathbb R^r$. For $X=(i\theta_1,\dots,i\theta_r)\in i\mathbb R^r$, the exponential map is taken coordinate by coordinate: \begin{align*} \exp(X)=(e^{i\theta_1},\dots,e^{i\theta_r}). \end{align*} This map is onto because every $z_j\in S^1$ can be written as $z_j=e^{i\theta_j}$ for some $\theta_j\in\mathbb R$. Now compute its kernel. We have $\exp(i\theta_1,\dots,i\theta_r)=(1,\dots,1)$ exactly when $e^{i\theta_j}=1$ for every $j$. For one coordinate, \begin{align*} e^{i\theta_j}=1 \quad\Longleftrightarrow\quad \theta_j\in 2\pi\mathbb Z. \end{align*} Therefore \begin{align*} \ker(\exp)=\{(2\pi i m_1,\dots,2\pi i m_r):m_j\in\mathbb Z\}=2\pi i\mathbb Z^r. \end{align*} Hence the exponential map identifies points of $i\mathbb R^r$ precisely when they differ by an element of $2\pi i\mathbb Z^r$, giving \begin{align*} T^r\cong i\mathbb R^r/(2\pi i\mathbb Z^r). \end{align*} Thus the standard torus is obtained from its Lie algebra by quotienting by the full lattice $2\pi i\mathbb Z^r$. [/example] The standard torus explains the abstract shape, but the course needs concrete subgroups inside the compact classical groups. The most important examples are diagonal unitary matrices, because later roots and weights will be read from their diagonal entries. [definition: Diagonal Torus in Unitary Groups] The diagonal torus in $U(n)$ is \begin{align*} T_{U(n)}=\{\operatorname{diag}(z_1,\dots,z_n):z_j\in S^1\}. \end{align*} The diagonal torus in $SU(n)$ is \begin{align*} T_{SU(n)}=\{\operatorname{diag}(z_1,\dots,z_n):z_j\in S^1,\ z_1\cdots z_n=1\}. \end{align*} [/definition] These groups have dimensions $n$ and $n-1$ respectively. Their Lie algebras are obtained by differentiating the defining equations, and this computation will be reused when the root system of $SU(n)$ is introduced. [example: Lie Algebras of the Diagonal Tori] For $U(n)$, a smooth curve in the diagonal torus has the form \begin{align*} \gamma(t)=\operatorname{diag}(z_1(t),\dots,z_n(t)), \end{align*} where each $z_j(t)\in S^1$ and $\gamma(0)=I$. Since $z_j(t)\in S^1$, we may write $z_j(t)=e^{i\theta_j(t)}$ near $t=0$ with $\theta_j(t)\in\mathbb R$ and $\theta_j(0)=0$. Differentiating at $t=0$ gives \begin{align*} z_j'(0)=\frac{d}{dt}\bigg|_{t=0}e^{i\theta_j(t)}=i\theta_j'(0)e^{i\theta_j(0)}=i\theta_j'(0). \end{align*} Thus \begin{align*} \gamma'(0)=\operatorname{diag}(i\theta_1'(0),\dots,i\theta_n'(0)). \end{align*} Conversely, for any [real numbers](/page/Real%20Numbers) $\theta_1,\dots,\theta_n$, the curve \begin{align*} \gamma(t)=\operatorname{diag}(e^{it\theta_1},\dots,e^{it\theta_n}) \end{align*} lies in $T_{U(n)}$, starts at $I$, and satisfies \begin{align*} \gamma'(0)=\operatorname{diag}(i\theta_1,\dots,i\theta_n). \end{align*} Therefore \begin{align*} \mathfrak t_{U(n)}=\{\operatorname{diag}(i\theta_1,\dots,i\theta_n):\theta_j\in\mathbb R\}. \end{align*} For $SU(n)$ we impose the extra condition $\det\gamma(t)=1$. For the diagonal curve above, \begin{align*} \det\gamma(t)=e^{it\theta_1}\cdots e^{it\theta_n}=e^{it(\theta_1+\cdots+\theta_n)}. \end{align*} This determinant is equal to $1$ for all $t$ exactly when $\theta_1+\cdots+\theta_n=0$. Equivalently, the tangent matrix has trace \begin{align*} \operatorname{tr}\operatorname{diag}(i\theta_1,\dots,i\theta_n)=i(\theta_1+\cdots+\theta_n), \end{align*} so the determinant-one condition becomes trace zero infinitesimally. Hence \begin{align*} \mathfrak t_{SU(n)}=\{\operatorname{diag}(i\theta_1,\dots,i\theta_n):\theta_j\in\mathbb R,\ \theta_1+\cdots+\theta_n=0\}. \end{align*} Finally, exponentiating one of these diagonal tangent matrices gives \begin{align*} \exp(\operatorname{diag}(i\theta_1,\dots,i\theta_n))=\operatorname{diag}(e^{i\theta_1},\dots,e^{i\theta_n}), \end{align*} and in the $SU(n)$ case the product of the diagonal entries is $e^{i(\theta_1+\cdots+\theta_n)}=1$. Thus the Lie algebra calculation exactly matches the corresponding diagonal unitary torus. [/example] The diagonal examples would be less useful if they depended on having chosen a special basis in advance. The next result says that any abelian unitary subgroup can be brought into this diagonal form, so the diagonal torus is the universal local model for compact abelian matrix subgroups. [quotetheorem:9717] [citeproof:9717] This theorem is the matrix-group shadow of the maximal torus theorem. It says that compact abelian subgroups of $U(n)$ do not hide non-diagonal behaviour after conjugation. The hypotheses matter: a commuting family of matrices that is not normal need not be simultaneously unitarily diagonalizable, and a compact subgroup of $U(n)$ that is not abelian, such as $SU(2)\le U(2)$, is not expected to lie in any diagonal torus. Thus the result is not a diagonalisation theorem for all compact subgroups; it is the precise abelian case. The same idea has a geometric incarnation in $SO(3)$, where diagonalisation becomes the choice of an axis. [example: Rotations About an Axis in SO(3)] Fix the $x_3$-axis, and let $R_\theta$ be the rotation through angle $\theta$ about this axis, so \begin{align*} R_\theta(a,b,c)=(a\cos\theta-b\sin\theta,\ a\sin\theta+b\cos\theta,\ c). \end{align*} The squared norm is preserved because \begin{align*} (a\cos\theta-b\sin\theta)^2+(a\sin\theta+b\cos\theta)^2+c^2=a^2+b^2+c^2. \end{align*} Its determinant is the determinant of the planar rotation on $e_3^\perp$ times the determinant on $\mathbb Re_3$, namely \begin{align*} (\cos^2\theta+\sin^2\theta)\cdot 1=1, \end{align*} so $R_\theta\in SO(3)$. Also $R_\theta e_3=e_3$, and the multiplication rule follows from the angle-addition identities: \begin{align*} R_\theta R_\phi(a,b,c)=R_{\theta+\phi}(a,b,c). \end{align*} Thus $T=\{R_\theta:\theta\in\mathbb R\}$ is a subgroup of $SO(3)$, and $e^{i\theta}\mapsto R_\theta$ gives a well-defined Lie group isomorphism $S^1\cong T$. To check maximality, let $H\le SO(3)$ be a connected abelian subgroup with $T\subseteq H$, and choose $h\in H$. Since $H$ is abelian, $hR_\theta=R_\theta h$ for every $\theta$. Applying both sides to $e_3$ gives \begin{align*} h e_3=hR_\theta e_3=R_\theta h e_3. \end{align*} Hence $h e_3$ is fixed by every $R_\theta$. If $v=(a,b,c)$ is fixed by $R_\pi$, then \begin{align*} R_\pi v=(-a,-b,c)=v, \end{align*} so $a=b=0$. Therefore the common fixed subspace of all rotations $R_\theta$ is $\mathbb Re_3$, and $h e_3=\lambda e_3$ for some $\lambda\in\mathbb R$. Since $h$ is orthogonal, $\|h e_3\|=\|e_3\|=1$, so $\lambda=\pm 1$. The sign $\lambda$ varies continuously with $h$, while $\{\pm1\}$ is discrete; because $H$ is connected and the identity has $\lambda=1$, every $h\in H$ satisfies $h e_3=e_3$. Thus every $h\in H$ fixes $e_3$ and preserves the orthogonal plane $e_3^\perp$. Since $\det(h)=1$ and $h$ acts as the identity on $\mathbb Re_3$, its restriction to $e_3^\perp$ is an element of $SO(2)$, hence is rotation through some angle $\phi$. Therefore $h=R_\phi\in T$. We have shown $H\subseteq T$, and the reverse inclusion was assumed, so $H=T$. Thus rotations about a fixed axis form a maximal torus in $SO(3)$: geometrically, it is the circle of all possible angles around that axis. [/example] This example already hints at conjugacy. Choosing a different axis gives a different copy of $S^1$, but a rotation carrying one axis to the other conjugates the corresponding subgroups. ## Maximal Tori and the Exponential Image of a Maximal Abelian Subalgebra The next problem is intrinsic: given a compact connected Lie group $G$, how do we find the torus without first placing $G$ inside a unitary group and diagonalising matrices? The Lie algebra gives the answer. Start with a maximal abelian subalgebra, exponentiate it, and then close up if needed. To state the group-level object, maximality must be imposed among tori rather than among all abelian subgroups. This keeps the focus on connected compact abelian Lie subgroups, which are the subgroups controlled by Lie algebra data. [definition: Maximal Torus] Let $G$ be a Lie group. A maximal torus in $G$ is a torus $T\le G$ that is not properly contained in any larger torus of $G$. [/definition] Maximality is among connected compact abelian subgroups, not among all abelian subgroups. To construct such a subgroup from the infinitesimal side, we need the corresponding maximality condition inside the Lie algebra. [definition: Maximal Abelian Subalgebra] Let $\mathfrak g$ be a Lie algebra. A subalgebra $\mathfrak t\le \mathfrak g$ is a maximal abelian subalgebra if $[X,Y]=0$ for all $X,Y\in\mathfrak t$ and no strictly larger Lie subalgebra of $\mathfrak g$ has this property. [/definition] The previous definition gives an infinitesimal candidate for the torus, but exponentiating a subspace of a compact group can produce a dense immersed subgroup rather than a closed subgroup. The obstruction is that Lie algebra data alone gives local commuting directions, while a maximal torus is a global compact subgroup. The needed bridge is a result saying that after taking closure, those infinitesimal directions become the maximal torus they were meant to describe. [quotetheorem:9718] [citeproof:9718] The closure in the theorem is not cosmetic. In a torus such as $(S^1)^2$, a one-parameter subgroup with irrational slope has dense image, so $\exp(\mathfrak a)$ can fail to be closed even though its closure is a torus. Compactness is what turns this closure into a compact Lie subgroup; in noncompact groups, exponentials of abelian subalgebras need not produce compact toral objects at all. The theorem therefore does not say that $\exp(\mathfrak t)$ itself is always embedded or closed. The preceding theorem is often used in the reverse direction: a maximal torus should have no missing commuting infinitesimal directions. Without that reverse statement, a group-level maximal torus could still have a Lie algebra sitting inside a larger abelian subalgebra, so infinitesimal calculations would not faithfully reflect maximality in the group. The reverse implication is what allows the course to pass between torus-level and Lie-algebra-level statements. [quotetheorem:9719] [citeproof:9719] These two theorems justify the common notation $\mathfrak t=\operatorname{Lie}(T)$ for a chosen maximal torus $T$. The maximality assumption is essential: a proper subtorus of a maximal torus has an abelian Lie algebra, but it misses commuting directions and therefore cannot control the group. Compact connectedness is also part of the mechanism; without compactness there may be maximal abelian subgroups that are not tori, and without connectedness different components can introduce finite commuting elements invisible to the Lie algebra. A new bookkeeping problem now appears: to compare different compact connected groups, we need a single number recording the size of their maximal abelian direction. That number is the rank. [definition: Rank of a Compact Connected Lie Group] Let $G$ be a compact connected Lie group. The rank of $G$ is the dimension of any maximal torus of $G$. [/definition] The definition is independent of the maximal torus chosen because maximal tori will be conjugate. Before proving that, the classical examples show what the rank measures. [example: Ranks of U(n), SU(n), and SO(3)] For $U(n)$, the diagonal torus is \begin{align*} T_{U(n)}=\{\operatorname{diag}(e^{i\theta_1},\dots,e^{i\theta_n}):\theta_1,\dots,\theta_n\in\mathbb R\}. \end{align*} Its parameters are independent modulo $2\pi\mathbb Z$ in each coordinate: changing $\theta_j$ changes only the $j$th diagonal entry. Thus \begin{align*} T_{U(n)}\cong (S^1)^n, \end{align*} so this maximal torus has dimension $n$, and therefore $\operatorname{rank}U(n)=n$. For $SU(n)$, the determinant-one condition imposes one relation on the same diagonal entries: \begin{align*} \det\operatorname{diag}(e^{i\theta_1},\dots,e^{i\theta_n})=e^{i\theta_1}\cdots e^{i\theta_n}=e^{i(\theta_1+\cdots+\theta_n)}. \end{align*} Infinitesimally, the diagonal tangent matrix $\operatorname{diag}(i\theta_1,\dots,i\theta_n)$ has trace \begin{align*} \operatorname{tr}\operatorname{diag}(i\theta_1,\dots,i\theta_n)=i\theta_1+\cdots+i\theta_n=i(\theta_1+\cdots+\theta_n). \end{align*} So the Lie algebra of the diagonal torus in $SU(n)$ is cut out by \begin{align*} \theta_1+\cdots+\theta_n=0. \end{align*} The first $n-1$ real numbers $\theta_1,\dots,\theta_{n-1}$ may be chosen freely, and then the last one is forced to be \begin{align*} \theta_n=-(\theta_1+\cdots+\theta_{n-1}). \end{align*} Hence the diagonal maximal torus in $SU(n)$ has dimension $n-1$, and $\operatorname{rank}SU(n)=n-1$. In $SO(3)$, the rotations about a fixed axis form the subgroup \begin{align*} T=\{R_\theta:\theta\in\mathbb R\}. \end{align*} The map $S^1\to T$ given by $e^{i\theta}\mapsto R_\theta$ is an isomorphism, so $T$ is a one-dimensional torus. Since the rotations-about-an-axis example showed that this circle is maximal in $SO(3)$, it follows that $\operatorname{rank}SO(3)=1$. Thus rank is computed by finding a maximal torus and counting its independent angle parameters. [/example] Even orthogonal groups display a second standard pattern: independent rotations in mutually orthogonal planes. [example: Block Rotations in SO(2n)] Write $\mathbb R^{2n}$ as the orthogonal direct sum of coordinate planes \begin{align*} \mathbb R^{2n}=P_1\oplus\cdots\oplus P_n,\qquad P_j=\operatorname{span}(e_{2j-1},e_{2j}). \end{align*} For angles $\theta_1,\dots,\theta_n\in\mathbb R$, define $R(\theta_1,\dots,\theta_n)$ by rotating $P_j$ through angle $\theta_j$: \begin{align*} R(\theta_1,\dots,\theta_n)(x_j,y_j)_{j=1}^n=(x_j\cos\theta_j-y_j\sin\theta_j,\ x_j\sin\theta_j+y_j\cos\theta_j)_{j=1}^n. \end{align*} On the $j$th plane, \begin{align*} (x_j\cos\theta_j-y_j\sin\theta_j)^2+(x_j\sin\theta_j+y_j\cos\theta_j)^2=x_j^2(\cos^2\theta_j+\sin^2\theta_j)+y_j^2(\sin^2\theta_j+\cos^2\theta_j). \end{align*} The mixed terms are $-2x_jy_j\cos\theta_j\sin\theta_j+2x_jy_j\sin\theta_j\cos\theta_j=0$, so this equals $x_j^2+y_j^2$. Summing over $j$ shows that $R(\theta_1,\dots,\theta_n)$ is orthogonal. Each planar rotation has determinant \begin{align*} \cos\theta_j\cdot\cos\theta_j-(-\sin\theta_j)\sin\theta_j=\cos^2\theta_j+\sin^2\theta_j=1, \end{align*} so the block diagonal determinant is $1\cdots 1=1$. Hence $R(\theta_1,\dots,\theta_n)\in SO(2n)$. The angle-addition identities give, on each plane, \begin{align*} R(\theta_1,\dots,\theta_n)R(\phi_1,\dots,\phi_n)=R(\theta_1+\phi_1,\dots,\theta_n+\phi_n). \end{align*} Thus \begin{align*} T=\{R(\theta_1,\dots,\theta_n):\theta_j\in\mathbb R\} \end{align*} is an abelian subgroup. The map $(S^1)^n\to T$ sending $(e^{i\theta_1},\dots,e^{i\theta_n})$ to $R(\theta_1,\dots,\theta_n)$ is well-defined because the formula only uses $\cos\theta_j$ and $\sin\theta_j$, and it is a Lie group isomorphism. Therefore $T$ is an $n$-dimensional torus. Its Lie algebra is obtained by differentiating the curve $t\mapsto R(t\alpha_1,\dots,t\alpha_n)$ at $t=0$. On the $j$th coordinate plane, \begin{align*} \frac{d}{dt}\bigg|_{t=0}(x_j\cos(t\alpha_j)-y_j\sin(t\alpha_j),\ x_j\sin(t\alpha_j)+y_j\cos(t\alpha_j))=(-\alpha_j y_j,\alpha_j x_j). \end{align*} So the tangent vectors are exactly the infinitesimal rotations with independent real speeds $\alpha_1,\dots,\alpha_n$ on the mutually orthogonal planes $P_1,\dots,P_n$. Finally, if a connected abelian subgroup $H\le SO(2n)$ contains $T$, then every $h\in H$ commutes with every element of $T$. For fixed $j$, choose angles with $\theta_j$ arbitrary and all other angles $0$. The operator $h$ must commute with every rotation of $P_j$ while also commuting with rotations on the other coordinate planes. The eigenspace decomposition over $\mathbb C$ for these independent rotations separates the complex lines attached to the $n$ planes, so $h$ preserves each real plane $P_j$. Since $h|_{P_j}\in O(2)$ and commutes with every rotation of $P_j$, it cannot be a reflection; hence $h|_{P_j}$ is itself a rotation. Therefore $h=R(\phi_1,\dots,\phi_n)\in T$. Thus $H\subseteq T$, and since $T\subseteq H$, we have $H=T$. Hence $T$ is a maximal torus of $SO(2n)$, and \begin{align*} \operatorname{rank}SO(2n)=\dim T=n. \end{align*} [/example] The examples suggest that a maximal torus is large but still much smaller than $G$. The next section explains why it nevertheless meets every conjugacy class. ## Conjugacy of Maximal Tori and Conjugacy Into a Maximal Torus The main question is how much of $G$ is visible from a fixed maximal torus $T$. The answer is stronger than density or genericity: every element of a compact connected Lie group is conjugate to an element of $T$, and every maximal torus is obtained from $T$ by conjugation. The first theorem is often called the maximal torus theorem. It is the compact Lie group analogue of diagonalising a unitary matrix. [quotetheorem:9713] [citeproof:9713] This theorem turns questions about conjugacy-invariant functions on $G$ into questions about functions on $T$ with finite symmetry. It does not say that an element has a unique representative in $T$; in $SU(n)$, for instance, diagonal matrices with permuted diagonal entries are conjugate. Compactness is also essential to the minimisation argument and to the conclusion: in a noncompact connected group such as $SL(2,\mathbb R)$, hyperbolic and unipotent elements cannot all be conjugated into one compact torus. Chapter 3 will identify the remaining finite ambiguity as the Weyl group. [example: Diagonalising Elements of SU(n)] Let $g\in SU(n)$. Since $g$ is unitary, the *Spectral Theorem for Unitary Matrices* gives a unitary matrix $u\in U(n)$ such that \begin{align*} d=ugu^{-1} \end{align*} is diagonal. We first replace $u$ by a conjugating matrix of determinant $1$. Since $u\in U(n)$, $\det u\in S^1$, so write $\det u=e^{i\alpha}$ for some $\alpha\in\mathbb R$. Set \begin{align*} \lambda=e^{-i\alpha/n}. \end{align*} Then $\lambda\in S^1$, and for $v=\lambda u$ we have \begin{align*} \det v=\det(\lambda u)=\lambda^n\det u=e^{-i\alpha}e^{i\alpha}=1. \end{align*} Thus $v\in SU(n)$. Because $\lambda I$ is scalar, it commutes with $g$, so \begin{align*} vgv^{-1}=(\lambda u)g(\lambda u)^{-1}=\lambda ugu^{-1}\lambda^{-1}=ugu^{-1}=d. \end{align*} It remains to check that this diagonal conjugate lies in $T_{SU(n)}$. Since $v,g\in SU(n)$, \begin{align*} \det(vgv^{-1})=\det v\cdot \det g\cdot \det(v^{-1})=1\cdot 1\cdot 1=1. \end{align*} Also $d$ is unitary because \begin{align*} d^*d=(vgv^{-1})^*(vgv^{-1})=vg^*v^{-1}vgv^{-1}=v(g^*g)v^{-1}=vv^{-1}=I. \end{align*} Writing $d=\operatorname{diag}(z_1,\dots,z_n)$, the equation $d^*d=I$ says \begin{align*} \operatorname{diag}(|z_1|^2,\dots,|z_n|^2)=I, \end{align*} so each $z_j\in S^1$. Together with $\det d=z_1\cdots z_n=1$, this gives $d\in T_{SU(n)}$. Hence every element of $SU(n)$ is conjugate, by an element of $SU(n)$ itself, to an element of the diagonal torus $T_{SU(n)}$. [/example] Conjugating elements into a fixed torus would be less coherent if different maximal tori led to unrelated pictures. The obstruction is not existence of tori but choice: a compact connected group may contain many maximal tori, and calculations on one of them would be arbitrary unless any other choice could be transported to it by an element of the group. The needed structural fact is that maximal tori form a single conjugacy class. [quotetheorem:9720] [citeproof:9720] This result is one of the first places where compactness and connectedness work together in the structure theory. Compactness supplies the invariant metric and fixed point input, while connectedness prevents the theory from splitting into unrelated component-wise statements. The conclusion can fail in disconnected compact groups: different components may contain maximal connected abelian subgroups that are not conjugate inside the full group. The theorem also does not identify a canonical maximal torus; it says only that the set of choices is a single conjugacy class. [example: Maximal Tori in SO(3)] Let $T\le SO(3)$ be a maximal torus. It is not the one-point subgroup, because the one-point subgroup is contained in the circle of rotations about any fixed axis. Choose a nonidentity element $r\in T$. Since $r\in SO(3)$ is a nontrivial rotation, its fixed subspace is an axis $\ell$ through the origin. For any $t\in T$, commutativity gives $tr=rt$, so if $v\in \ell$ then $rv=v$ and therefore \begin{align*} r(tv)=t(rv)=tv. \end{align*} Thus $tv\in \ell$, so every $t\in T$ preserves $\ell$. Because $T$ is connected and the action on the line $\ell$ has values in the discrete group $\{\pm 1\}$, every element of $T$ acts on $\ell$ with the same sign as the identity, namely $+1$. Hence every $t\in T$ fixes $\ell$ pointwise and acts on $\ell^\perp$ as an element of $SO(2)$, so $t$ is a rotation about $\ell$. Maximality then forces $T$ to be the full circle of rotations about $\ell$, since the full rotation circle about $\ell$ is a torus containing $T$. Now let $T_1$ and $T_2$ be maximal tori corresponding to axes $\ell_1$ and $\ell_2$. Choose unit vectors $u_j\in \ell_j$, and choose $g\in SO(3)$ with $g u_1=u_2$ by sending an oriented orthonormal basis beginning with $u_1$ to one beginning with $u_2$. If $R\in T_1$, then $Ru_1=u_1$, so \begin{align*} (gRg^{-1})u_2=(gRg^{-1})(g u_1)=gRu_1=gu_1=u_2. \end{align*} Thus $gRg^{-1}$ fixes $\ell_2$ and is therefore a rotation about $\ell_2$, so $gT_1g^{-1}\subseteq T_2$. Applying the same argument to $g^{-1}$ gives $T_2\subseteq gT_1g^{-1}$, hence \begin{align*} gT_1g^{-1}=T_2. \end{align*} Thus conjugating maximal tori in $SO(3)$ is exactly the geometric operation of rotating one axis onto another. [/example] The conjugacy theorems tell us that a maximal torus is the right abelian subgroup to study. To measure the residual symmetry of this torus, we need to distinguish elements that fix every point of $T$ from elements that merely preserve $T$ as a subgroup. [definition: Centralizer and Normalizer] Let $G$ be a group and let $H\le G$. The centralizer of $H$ in $G$ is \begin{align*} C_G(H)=\{g\in G:gh=hg\text{ for all }h\in H\}. \end{align*} The normalizer of $H$ in $G$ is \begin{align*} N_G(H)=\{g\in G:gHg^{-1}=H\}. \end{align*} [/definition] The centralizer consists of elements that commute with the whole torus, whereas the normalizer permits conjugation symmetries of the torus. For a maximal torus in a compact connected group, there are no extra elements centralizing it, and this fact is what makes the later quotient $N_G(T)/T$ faithful to the remaining finite symmetry. [quotetheorem:9721] [citeproof:9721] This connectedness theorem is exactly the point that prevents a finite centralizing component from surviving. The Lie algebra can only see the identity component, so a separate connectedness input is needed before an infinitesimal centralizer calculation can imply a group-level equality. The remaining issue is whether the connected centralizer is merely connected or actually no larger than the maximal torus itself. Maximality should rule out any extra commuting directions, and that is the group-level equality needed before the quotient $N_G(T)/T$ can be interpreted as the nontrivial symmetry left over after centralizing elements have been removed. [quotetheorem:9722] [citeproof:9722] This result separates two different symmetries of $T$. Elements that centralize $T$ are already in $T$, while elements that normalize $T$ may act nontrivially on it. The statement uses connectedness: in a disconnected compact group, a finite component can centralize the identity component of a torus without lying in that torus. The theorem also depends on maximality; a proper subtorus of $T$ has a larger centralizer containing at least all of $T$. The quotient $N_G(T)/T$ is therefore finite; Chapter 3 names this quotient the Weyl group and studies its action on $T$ and on $\mathfrak t$. [remark: Why the Normalizer Matters] If $g\in N_G(T)$, then conjugation by $g$ defines an automorphism of $T$. If $g\in T$, this automorphism is the identity because $T$ is abelian. Thus the nontrivial action on $T$ is measured by the quotient $N_G(T)/T$, not by $N_G(T)$ itself. [/remark] The chapter has shown that maximal tori are both abundant and unique up to conjugation. Abundance means every group element is represented inside a torus; uniqueness means that computations made on a chosen torus do not depend on an arbitrary choice except for the finite ambiguity coming from the normalizer. The rest of the course turns that finite ambiguity into the Weyl group and then uses it to formulate the root, weight, integration, and character theories of compact Lie groups. The previous chapter showed that maximal tori meet every conjugacy class, so many problems reduce to understanding what happens inside a torus. What remains is the finite ambiguity coming from conjugation within the torus itself, and that ambiguity is encoded by the Weyl group and its chambers. # 3. Weyl Groups and Weyl Chambers The preceding chapter showed that maximal tori are large enough to meet every conjugacy class in a compact connected Lie group. This chapter studies the remaining ambiguity: an element of a maximal torus may be conjugate to other elements of the same torus. The finite group measuring that ambiguity is the Weyl group, and its linear action cuts the Lie algebra of the torus into Weyl chambers. These chambers give a geometric fundamental domain for conjugacy and prepare the root-theoretic structure developed later in the course. ## Normalizers and the Weyl Group Once a maximal torus $T$ has been chosen, the central question is not whether every element of $G$ can be moved into $T$, but how much freedom remains after it has been moved there. If $g t g^{-1}$ lies in $T$ for every $t \in T$, then conjugation by $g$ produces an automorphism of $T$. The subgroup of all such elements is the normalizer. [definition: Normalizer of a Maximal Torus] Let $G$ be a Lie group and let $T \le G$ be a maximal torus. The normalizer of $T$ in $G$ is \begin{align*} N_G(T) := \{g \in G : gTg^{-1}=T\}. \end{align*} [/definition] The normalizer tells us which conjugations preserve the chosen torus, but it still contains the whole torus itself. Since elements of $T$ act as the identity on $T$ by conjugation, we need to quotient out this redundant continuous part to isolate the residual finite symmetry. [definition: Weyl Group] Let $G$ be a compact connected Lie group and let $T \le G$ be a maximal torus. The Weyl group of the pair $(G,T)$ is \begin{align*} W(G,T) := N_G(T)/T. \end{align*} When the maximal torus is fixed, write $W$ for $W(G,T)$. [/definition] The quotient is a group because $T$ is a [normal subgroup](/page/Normal%20Subgroup) of $N_G(T)$. Since all maximal tori are conjugate, different choices of $T$ give conjugate versions of the same finite symmetry group, so the notation $W(G)$ is often used when no confusion is possible. [remark: Weyl Group as Residual Conjugacy] An element $n \in N_G(T)$ acts on $T$ by $t \mapsto ntn^{-1}$. Elements of $T$ act as the identity on $T$ because $T$ is abelian, so this action descends to an action of $W=N_G(T)/T$ on $T$. [/remark] The preceding remark turns the quotient into an actual symmetry of the torus. To compare this symmetry with the linear geometry of roots and chambers, we next differentiate the action from $T$ to its Lie algebra $\mathfrak t$. [definition: Weyl Group Action on the Torus Lie Algebra] Let $G$ be a compact connected Lie group, let $T\le G$ be a maximal torus, and let $W=N_G(T)/T$. For $n\in N_G(T)$, define the conjugation map \begin{align*} c_n:G&\to G, & c_n(g)&=ngn^{-1}. \end{align*} For $w=nT\in W$, the induced action on $T$ is the map \begin{align*} W\times T&\to T, & (w,t)&\mapsto w\cdot t:=c_n(t)=ntn^{-1}. \end{align*} The induced action on $\mathfrak t$ is the map \begin{align*} W\times \mathfrak t&\to \mathfrak t, & (w,X)&\mapsto w\cdot X:=d(c_n)_e(X). \end{align*} [/definition] This is well-defined on cosets: replacing $n$ by $nt_0$ with $t_0\in T$ does not change the induced action on $T$ or on $\mathfrak t$. The compatibility with the exponential map is \begin{align*} \exp(w\cdot X)=w\cdot \exp(X), \qquad X\in \mathfrak t. \end{align*} For the chamber picture and for later character formulae, the action must be controlled by a finite group rather than by a positive-dimensional family; this is the next structural fact. [quotetheorem:9723] [citeproof:9723] The theorem is not saying that every normalizer quotient in Lie theory is finite; it is using compactness and maximality at the same time. If compactness is dropped, the additive Lie group $\mathbb R$ has maximal torus $\{0\}$, so $N_{\mathbb R}(\{0\})/\{0\}\cong \mathbb R$ is infinite. If maximality is dropped inside a compact connected group, for instance taking $T=\{I\}$ in $SU(2)$, the quotient $N_G(T)/T$ is the whole positive-dimensional group $SU(2)$ rather than a finite reflection group. The result therefore isolates the finite residual symmetry only after the torus is as large as possible in a compact connected group. [example: The Weyl Group of SU(n)] Let $G=SU(n)$ and let \begin{align*}T=\{\operatorname{diag}(z_1,\dots,z_n): |z_i|=1,\ z_1\cdots z_n=1\}.\end{align*} For $\sigma\in S_n$, let $P_\sigma$ be the permutation matrix satisfying $P_\sigma e_j=e_{\sigma(j)}$. If $t=\operatorname{diag}(z_1,\dots,z_n)$, then \begin{align*}P_\sigma tP_\sigma^{-1}e_i=P_\sigma t e_{\sigma^{-1}(i)}=P_\sigma(z_{\sigma^{-1}(i)}e_{\sigma^{-1}(i)})=z_{\sigma^{-1}(i)}e_i.\end{align*} Hence \begin{align*}P_\sigma tP_\sigma^{-1}=\operatorname{diag}(z_{\sigma^{-1}(1)},\dots,z_{\sigma^{-1}(n)}),\end{align*} so $P_\sigma$ normalizes the diagonal torus inside $U(n)$. If $\det(P_\sigma)\ne 1$, choose \begin{align*}d_\sigma=\operatorname{diag}(\det(P_\sigma)^{-1},1,\dots,1)\in T.\end{align*} Then $d_\sigma P_\sigma\in SU(n)$ because \begin{align*}\det(d_\sigma P_\sigma)=\det(d_\sigma)\det(P_\sigma)=\det(P_\sigma)^{-1}\det(P_\sigma)=1.\end{align*} Since $d_\sigma$ is diagonal, it commutes with every $t\in T$, and therefore $d_\sigma P_\sigma$ induces the same conjugation action on $T$ as $P_\sigma$. Conversely, let $n=(n_{ij})\in N_{SU(n)}(T)$. Choose $t=\operatorname{diag}(z_1,\dots,z_n)\in T$ with the $z_j$ pairwise distinct, and write \begin{align*}ntn^{-1}=t'=\operatorname{diag}(z'_1,\dots,z'_n).\end{align*} The equation $nt=t'n$ gives, in the $(i,j)$ entry, \begin{align*}n_{ij}z_j=z'_i n_{ij}.\end{align*} Thus if $n_{ij}\ne 0$, then $z_j=z'_i$. Because the $z_j$ are pairwise distinct and $n$ is unitary, each row and each column has exactly one nonzero entry. Hence $n$ is a diagonal unitary matrix times a permutation matrix. Modulo the diagonal subgroup $T$, only the permutation remains, so \begin{align*}W(SU(n),T)\cong S_n.\end{align*} The action is \begin{align*}\sigma\cdot \operatorname{diag}(z_1,\dots,z_n)=\operatorname{diag}(z_{\sigma^{-1}(1)},\dots,z_{\sigma^{-1}(n)}).\end{align*} On \begin{align*}\mathfrak t=\{\operatorname{diag}(ix_1,\dots,ix_n): x_i\in\mathbb R,\ \sum_i x_i=0\},\end{align*} the same calculation gives \begin{align*}P_\sigma \operatorname{diag}(ix_1,\dots,ix_n)P_\sigma^{-1}=\operatorname{diag}(ix_{\sigma^{-1}(1)},\dots,ix_{\sigma^{-1}(n)}).\end{align*} Thus the Weyl group records exactly the freedom to reorder the diagonal entries, and the determinant correction lies inside $T$ so it disappears in the quotient. [/example] This example is the prototype: a compact matrix group has a diagonal or block-diagonal torus, and the Weyl group consists of the allowed coordinate symmetries which preserve the form defining the group. It also shows why the quotient by $T$ in the definition of $W$ is essential: diagonal conjugations preserve the diagonal torus but do not change any diagonal entry, whereas permutation representatives genuinely move points of $T$. The next example is the rank-one version of the same phenomenon. Instead of many coordinates being permuted, there is only one angular coordinate, and the residual ambiguity is the sign change coming from reversing the chosen rotation axis. [example: The Weyl Group of SO(3)] Let $G=SO(3)$, and let $T=\{R_\theta:\theta\in\mathbb R\}$, where $R_\theta$ is the rotation about the $z$-axis given by \begin{align*}R_\theta e_1=\cos\theta\,e_1+\sin\theta\,e_2,\quad R_\theta e_2=-\sin\theta\,e_1+\cos\theta\,e_2,\quad R_\theta e_3=e_3.\end{align*} If $A\in N_{SO(3)}(T)$, then $AR_\theta A^{-1}\in T$ for every $\theta$. Choose $\theta$ with $\theta\not\equiv 0\pmod{\pi}$. The fixed subspace of $R_\theta$ is exactly $\mathbb R e_3$: if $v=ae_1+be_2+ce_3$ and $R_\theta v=v$, then \begin{align*}a\cos\theta-b\sin\theta=a,\quad a\sin\theta+b\cos\theta=b.\end{align*} Equivalently, \begin{align*}a(\cos\theta-1)-b\sin\theta=0,\quad a\sin\theta+b(\cos\theta-1)=0.\end{align*} Multiplying the first equation by $a$, the second by $b$, and adding gives \begin{align*}(a^2+b^2)(\cos\theta-1)=0.\end{align*} Since $\theta\not\equiv 0\pmod{2\pi}$, we get $a=b=0$, so $v\in\mathbb R e_3$. Therefore the fixed subspace of $AR_\theta A^{-1}$ is $A(\mathbb R e_3)$, but $AR_\theta A^{-1}\in T$ has fixed subspace $\mathbb R e_3$, so $A(\mathbb R e_3)=\mathbb R e_3$. Thus every normalizing rotation preserves the $z$-axis as an unoriented line. Conversely, any rotation preserving $\mathbb R e_3$ normalizes $T$. If it preserves the orientation of the axis, it lies in $T$. If it reverses the axis, it lies in $sT$, where $s$ is the rotation by $\pi$ about the $x$-axis: \begin{align*}s e_1=e_1,\quad s e_2=-e_2,\quad s e_3=-e_3.\end{align*} Since $s^{-1}=s$, we compute \begin{align*}sR_\theta s^{-1}e_1=sR_\theta e_1=s(\cos\theta\,e_1+\sin\theta\,e_2)=\cos\theta\,e_1-\sin\theta\,e_2=R_{-\theta}e_1.\end{align*} Also, \begin{align*}sR_\theta s^{-1}e_2=sR_\theta(-e_2)=s(\sin\theta\,e_1-\cos\theta\,e_2)=\sin\theta\,e_1+\cos\theta\,e_2=R_{-\theta}e_2.\end{align*} Finally, \begin{align*}sR_\theta s^{-1}e_3=sR_\theta(-e_3)=s(-e_3)=e_3=R_{-\theta}e_3.\end{align*} Hence $sR_\theta s^{-1}=R_{-\theta}$, so the nontrivial coset in $N_{SO(3)}(T)/T$ acts by angle reversal. Therefore \begin{align*}W(SO(3),T)\cong \mathbb Z/2\mathbb Z.\end{align*} On the Lie algebra $\mathfrak t\cong \mathbb R$, where $X$ represents infinitesimal rotation angle, this nontrivial element acts by $X\mapsto -X$. [/example] The rank-one case already shows the chamber picture: the line $\mathfrak t$ is divided by the singular point $0$ into two open rays. Higher-rank groups replace this single wall by a finite hyperplane arrangement. ## Regular Elements, Singular Hyperplanes, and Weyl Chambers Which points of $T$ or $\mathfrak t$ have the least residual symmetry? The answer is governed by centralizers. A point is regular when its centralizer is as small as possible, namely a maximal torus, and singular when extra symmetries appear. [definition: Regular Element of a Compact Lie Group] Let $G$ be a compact connected Lie group. An element $g\in G$ is regular if its centralizer \begin{align*} Z_G(g):=\{h\in G:hg=gh\} \end{align*} has a maximal torus as its identity component. [/definition] For elements already lying in the fixed maximal torus $T$, regularity means that no extra Lie algebra directions outside $\mathfrak t$ commute with the element. The root-space decomposition in Chapter 4 will make this criterion explicit by detecting the extra directions outside $\mathfrak t$ that commute with a torus element. For now, we need the geometric loci where the Weyl action has extra fixed directions; these loci are the singular hyperplanes. [definition: Singular Hyperplane] Let $T\le G$ be a maximal torus with Lie algebra $\mathfrak t$. A reflection of $\mathfrak t$ is a linear map $r:\mathfrak t\to \mathfrak t$ such that $r^2=\operatorname{id}_{\mathfrak t}$ and the fixed-point subspace $\ker(r-\operatorname{id}_{\mathfrak t})$ has codimension one. A singular hyperplane in $\mathfrak t$ is a codimension-one linear subspace fixed pointwise by a non-identity reflection arising from the Weyl group action. [/definition] The singular hyperplanes are the walls where stabilizers jump. To use them as a substitute for choosing ordered eigenvalues, we pass from the walls themselves to the open regions left after removing all walls. [definition: Weyl Chamber] Let $\mathcal H$ be the union of all singular hyperplanes in $\mathfrak t$. A Weyl chamber is a [connected component](/page/Connected%20Component) of \begin{align*} \mathfrak t \setminus \mathcal H. \end{align*} [/definition] A chamber is open in $\mathfrak t$ and its boundary lies in the union of singular hyperplanes. The next question is whether these regions form a manageable fundamental-domain system, rather than an arbitrary collection of open sets. [quotetheorem:4704] [citeproof:4704] This theorem packages a large amount of structure into a geometric statement, but it relies on the Weyl group being a reflection group, not merely on its finiteness. A general finite subgroup of $GL(V)$ need not have enough reflections to cut $V$ into chambers; for example, a non-trivial finite group acting on a real line by the identity has no reflecting walls and no simply transitive chamber action. The theorem is also a linearized statement on $\mathfrak t$: it describes the unfolded chamber picture before quotienting by the exponential lattice, so it is not yet a full description of the global quotient $T/W$. Later, roots supply the hyperplanes, weights provide the integral lattice data, and character formulae use both pieces at once. [example: Chambers for SU(n)] For $G=SU(n)$, write an element of the Lie algebra of the diagonal maximal torus as \begin{align*} \operatorname{diag}(ix_1,\dots,ix_n) \end{align*} with $x_i\in\mathbb R$ and trace condition \begin{align*} ix_1+\cdots+ix_n=i(x_1+\cdots+x_n)=0. \end{align*} Thus we identify \begin{align*} \mathfrak t\cong \{(x_1,\dots,x_n)\in \mathbb R^n: x_1+\cdots+x_n=0\}. \end{align*} From the computation of the Weyl group of $SU(n)$, a permutation $\sigma\in S_n$ acts by \begin{align*} \sigma\cdot (x_1,\dots,x_n)=(x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(n)}). \end{align*} In particular, the transposition $(ij)$ fixes exactly those points whose $i$th and $j$th coordinates agree: if $(ij)\cdot x=x$, then the $i$th coordinate gives $x_j=x_i$, and if $x_i=x_j$, then swapping the two coordinates leaves every coordinate unchanged. Hence the singular hyperplanes are \begin{align*} x_i=x_j,\qquad i\ne j. \end{align*} Removing these hyperplanes means requiring all coordinates to be pairwise distinct. On any connected component, the signs of all differences $x_i-x_j$ cannot change, because changing the sign of $x_i-x_j$ would force $x_i-x_j=0$ at some intermediate point. Therefore a chamber is determined by one strict ordering of the coordinates. One such chamber is \begin{align*} x_1>x_2>\cdots>x_n \end{align*} inside the trace-zero hyperplane $x_1+\cdots+x_n=0$. Its closure is obtained by allowing equality in the defining inequalities, so it is \begin{align*} \{(x_1,\dots,x_n): x_1\ge x_2\ge \cdots \ge x_n,\ x_1+\cdots+x_n=0\}. \end{align*} Thus the chambers for $SU(n)$ are exactly the regions where the diagonal entries have a fixed strict order. [/example] The inequalities in the example are the first appearance of dominance conditions. In representation theory, the same chamber will select the dominant weights of $SU(n)$. [example: Signed Permutations for SO(2n+1) and Sp(n)] For $SO(2n+1)$, take the standard maximal torus \begin{align*} T=\{\operatorname{diag}(R_{\theta_1},\dots,R_{\theta_n},1):\theta_i\in\mathbb R\}, \end{align*} where $R_{\theta_i}$ is rotation in the $i$th coordinate $2$-plane. A permutation $\sigma\in S_n$ is represented by the [orthogonal matrix](/page/Orthogonal%20Matrix) which sends the $j$th coordinate $2$-plane to the $\sigma(j)$th coordinate $2$-plane and fixes the last coordinate. Since interchanging two real $2$-planes is the product of two ordinary coordinate transpositions, it has determinant $1$, so these block-permutation matrices lie in $SO(2n+1)$ and conjugate \begin{align*} \operatorname{diag}(R_{\theta_1},\dots,R_{\theta_n},1) \end{align*} to \begin{align*} \operatorname{diag}(R_{\theta_{\sigma^{-1}(1)}},\dots,R_{\theta_{\sigma^{-1}(n)}},1). \end{align*} For a sign change in the $i$th coordinate, let $s_i$ reflect one axis in the $i$th coordinate $2$-plane and also send the final basis vector to its negative. Then $\det(s_i)=(-1)(-1)=1$. On the chosen $2$-plane, write the basis as $u_i,v_i$, with \begin{align*} R_{\theta_i}u_i=\cos\theta_i\,u_i+\sin\theta_i\,v_i \end{align*} and \begin{align*} R_{\theta_i}v_i=-\sin\theta_i\,u_i+\cos\theta_i\,v_i. \end{align*} If $s_i u_i=u_i$ and $s_i v_i=-v_i$, then \begin{align*} s_iR_{\theta_i}s_i^{-1}u_i=s_iR_{\theta_i}u_i=s_i(\cos\theta_i\,u_i+\sin\theta_i\,v_i)=\cos\theta_i\,u_i-\sin\theta_i\,v_i=R_{-\theta_i}u_i. \end{align*} Also, \begin{align*} s_iR_{\theta_i}s_i^{-1}v_i=s_iR_{\theta_i}(-v_i)=s_i(\sin\theta_i\,u_i-\cos\theta_i\,v_i)=\sin\theta_i\,u_i+\cos\theta_i\,v_i=R_{-\theta_i}v_i. \end{align*} Thus $s_i$ changes $\theta_i$ to $-\theta_i$ and fixes the other angles. For $Sp(n)$, using quaternionic coordinates, the standard maximal torus is \begin{align*} T=\{\operatorname{diag}(e^{i\theta_1},\dots,e^{i\theta_n}):\theta_i\in\mathbb R\}. \end{align*} Quaternionic permutation matrices permute the diagonal entries. Multiplication in the $i$th coordinate by $j$ changes the sign of the $i$th angle because $j^{-1}=-j$, $ji=-ij$, and therefore \begin{align*} j e^{i\theta_i} j^{-1}=j(\cos\theta_i+i\sin\theta_i)(-j)=\cos\theta_i-i\sin\theta_i=e^{-i\theta_i}. \end{align*} Thus both groups contain normalizer representatives acting by \begin{align*} (\theta_1,\dots,\theta_n)\mapsto (\varepsilon_1\theta_{\sigma^{-1}(1)},\dots,\varepsilon_n\theta_{\sigma^{-1}(n)}), \end{align*} where $\sigma\in S_n$ and each $\varepsilon_i\in\{\pm1\}$. Conversely, for a torus element with all $\theta_i$ nonzero and no two angles equal up to sign, the complex eigenvalues in the $SO(2n+1)$ model are $e^{\pm i\theta_1},\dots,e^{\pm i\theta_n},1$. The corresponding real rotation planes are distinguished by the unordered eigenvalue pairs $\{e^{i\theta_i},e^{-i\theta_i}\}$. Any normalizer element must preserve this eigenspace decomposition, so it can only permute the $2$-planes and possibly interchange $e^{i\theta_i}$ with $e^{-i\theta_i}$ inside each pair. The same argument in the quaternionic diagonal model for $Sp(n)$ says that the normalizer can only permute the quaternionic coordinate lines and replace $e^{i\theta_i}$ by $e^{-i\theta_i}$. Therefore \begin{align*} W\cong (\mathbb Z/2\mathbb Z)^n\rtimes S_n. \end{align*} There are $2^n$ independent sign choices and $n!$ coordinate permutations, so \begin{align*} |W|=2^n n!. \end{align*} The reflecting walls for this signed-permutation action are the hyperplanes $\theta_i=0$ and $\theta_i=\pm\theta_j$. Away from these walls, the absolute values $|\theta_i|$ are nonzero and pairwise distinct. First choose signs so that all coordinates become positive, and then apply a unique permutation placing them in decreasing order. This gives the unique representative satisfying \begin{align*} \theta_1>\theta_2>\cdots>\theta_n>0. \end{align*} This region is the standard Weyl chamber: it orders the rotation angles by size after the sign symmetries have made them positive. [/example] These signed permutation examples explain why Weyl groups are often reflection groups rather than merely permutation groups. The extra sign changes encode rotations in mutually orthogonal two-planes or quaternionic coordinate directions. ## Conjugacy Classes and the Quotient T/W The maximal torus theorem says that every conjugacy class meets $T$. The remaining question is whether two elements of $T$ which are conjugate in $G$ must be related by the Weyl group. The answer is yes, and it is the main reason the Weyl group is the correct symmetry group. [quotetheorem:9724] [citeproof:9724] This theorem gives the exact intersection of a conjugacy class with $T$, but only under the compact connected maximal-torus hypotheses. If $T$ is not maximal, the statement can fail in the most basic way: taking $T=\{I\}$ in $SU(2)$, elements outside $T$ are still conjugate into some maximal torus, but the chosen $T$ cannot record their conjugacy data. Connectedness is also part of the control mechanism; in disconnected compact groups, components may act on a torus by extra automorphisms not captured by the Weyl group of the identity component alone. The result does not classify all conjugacy classes until it is combined with the maximal torus theorem from Chapter 2, which says that every element of $G$ is conjugate into $T$; this is exactly what the next theorem does. [quotetheorem:9725] [citeproof:9725] The quotient $T/W$ should be read geometrically as a folded torus. The theorem is a classification of conjugacy classes as a set; it does not by itself describe the topology, smooth stratification, or measure factors on the quotient. The assumptions are again essential: in the noncompact connected group $SL(2,\mathbb R)$, not every element is conjugate into a compact torus, so compact-torus data cannot classify all conjugacy classes. Passing to $\mathfrak t$ through the exponential map gives a local picture in which the Weyl chambers are the unfolded fundamental regions before the integral lattice is imposed, while the global quotient still remembers periodicity in $T$. [example: Conjugacy Classes in SU(n)] Let $A\in SU(n)$. Since $A$ is unitary, the *Spectral Theorem for Normal Matrices* gives a unitary matrix $U\in U(n)$ and complex numbers $z_1,\dots,z_n$ such that \begin{align*}UAU^{-1}=\operatorname{diag}(z_1,\dots,z_n).\end{align*} For each eigenvalue, unitarity gives $|z_i|=1$. Taking determinants in the displayed equation gives \begin{align*}\det(A)=\det(U^{-1})\det(\operatorname{diag}(z_1,\dots,z_n))\det(U)=z_1\cdots z_n.\end{align*} Since $A\in SU(n)$, $\det(A)=1$, so $z_1\cdots z_n=1$. The diagonalizing matrix can be chosen in $SU(n)$. Let $c=\det(U)$ and put \begin{align*}q=\operatorname{diag}(c^{-1},1,\dots,1).\end{align*} Then $\det(q)=c^{-1}$, so \begin{align*}\det(qU)=\det(q)\det(U)=c^{-1}c=1.\end{align*} Because $q$ is diagonal, it commutes with $\operatorname{diag}(z_1,\dots,z_n)$, and therefore \begin{align*}(qU)A(qU)^{-1}=q(UAU^{-1})q^{-1}=q\operatorname{diag}(z_1,\dots,z_n)q^{-1}=\operatorname{diag}(z_1,\dots,z_n).\end{align*} Thus every element of $SU(n)$ is conjugate, by an element of $SU(n)$, to a point of the diagonal torus \begin{align*}T=\{\operatorname{diag}(z_1,\dots,z_n): |z_i|=1,\ z_1\cdots z_n=1\}.\end{align*} Now let \begin{align*}D=\operatorname{diag}(z_1,\dots,z_n)\end{align*} and \begin{align*}E=\operatorname{diag}(w_1,\dots,w_n)\end{align*} be two elements of $T$. If $E=SDS^{-1}$ for some $S\in SU(n)$, then $D$ and $E$ have the same [characteristic polynomial](/page/Characteristic%20Polynomial): \begin{align*}\det(\lambda I-E)=\det(\lambda I-SDS^{-1})=\det(S(\lambda I-D)S^{-1})=\det(\lambda I-D).\end{align*} Since \begin{align*}\det(\lambda I-D)=(\lambda-z_1)\cdots(\lambda-z_n)\end{align*} and \begin{align*}\det(\lambda I-E)=(\lambda-w_1)\cdots(\lambda-w_n),\end{align*} the multisets $\{z_1,\dots,z_n\}$ and $\{w_1,\dots,w_n\}$ are equal. Conversely, if these multisets are equal, choose $\sigma\in S_n$ such that \begin{align*}w_i=z_{\sigma^{-1}(i)}\end{align*} for every $i$. The permutation matrix $P_\sigma$ satisfies \begin{align*}P_\sigma D P_\sigma^{-1}=\operatorname{diag}(z_{\sigma^{-1}(1)},\dots,z_{\sigma^{-1}(n)})=E.\end{align*} If $\det(P_\sigma)\ne 1$, set \begin{align*}d_\sigma=\operatorname{diag}(\det(P_\sigma)^{-1},1,\dots,1).\end{align*} Then $d_\sigma\in T$, $\det(d_\sigma P_\sigma)=1$, and $d_\sigma$ commutes with $D$, so \begin{align*}(d_\sigma P_\sigma)D(d_\sigma P_\sigma)^{-1}=d_\sigma(P_\sigma D P_\sigma^{-1})d_\sigma^{-1}=d_\sigma E d_\sigma^{-1}=E.\end{align*} Therefore two elements of the diagonal torus are conjugate in $SU(n)$ exactly when their diagonal entries differ by a permutation. Conjugacy classes in $SU(n)$ are consequently parametrized by unordered $n$-tuples $(z_1,\dots,z_n)$ with $|z_i|=1$ and $z_1\cdots z_n=1$, which is exactly the quotient of $T$ by the $S_n$-action permuting the diagonal entries. [/example] The example shows why $T/W$ is a compact replacement for the set of eigenvalue data. The torus supplies ordered diagonal data, and the Weyl group removes the ordering. [remark: Regular Classes and Chamber Interiors] Regular conjugacy classes correspond to points of $T$ whose infinitesimal representatives avoid the singular walls, up to the exponential lattice. Singular classes occur on the images of the walls, where the centralizer is larger than a maximal torus. [/remark] This distinction between regular and singular classes becomes important in integration. The Weyl integration formula weights the maximal torus by a Jacobian which vanishes along the singular locus, reflecting the fact that conjugacy classes change dimension there. [example: The Rank-One Quotient for SO(3)] For $SO(3)$, fix the maximal torus $T=\{R_\theta:\theta\in\mathbb R\}$ of rotations about the $z$-axis. The angle coordinate is periodic: $R_\theta=R_\phi$ exactly when $\theta-\phi\in 2\pi\mathbb Z$, because comparing the image of $e_1$ gives $\cos\theta=\cos\phi$ and $\sin\theta=\sin\phi$. From the Weyl group computation for $SO(3)$, the nontrivial Weyl element sends \begin{align*} R_\theta\mapsto R_{-\theta}. \end{align*} Thus the quotient identifies two angles when \begin{align*} \phi\equiv \theta \pmod{2\pi} \end{align*} or when \begin{align*} \phi\equiv -\theta \pmod{2\pi}. \end{align*} Given any angle $\theta$, choose the unique $\beta\in[0,2\pi)$ with $\theta\equiv \beta\pmod{2\pi}$. If $0\le \beta\le \pi$, its representative in the quotient interval is $\beta$. If $\pi<\beta<2\pi$, then $-\beta\equiv 2\pi-\beta\pmod{2\pi}$, and $2\pi-\beta\in(0,\pi)$. Hence every Weyl orbit has a representative with \begin{align*} 0\le \theta\le \pi. \end{align*} The endpoints are precisely the fixed points of the sign symmetry on the circle of angles. At $\theta=0$, \begin{align*} R_0=\operatorname{id}. \end{align*} At $\theta=\pi$, \begin{align*} -\pi\equiv \pi \pmod{2\pi}, \end{align*} so the nontrivial Weyl element fixes the torus element $R_\pi$. Therefore conjugacy classes in $SO(3)$ are parametrized by the folded interval $0\le\theta\le\pi$, with the two endpoints recording the angles where the quotient has nontrivial Weyl stabilizer. [/example] The chapter's main message is that compact connected conjugacy theory is torus theory plus Weyl group symmetry. Maximal tori provide existence of representatives, the Weyl group records exactly the residual conjugacy inside the torus, and Weyl chambers give a linear fundamental-domain picture that will become the language of roots and weights. Weyl chambers organize the residual torus symmetry into a finite reflection picture. The next chapter explains how that reflection action arises from the adjoint representation and its root decomposition, turning the geometry of chambers into the algebra of roots. # 4. Roots and the Adjoint Representation Chapters 2 and 3 reduced much of the structure of a compact connected Lie group $G$ to a maximal torus $T$ and the finite Weyl group $W=N_G(T)/T$. This chapter explains the root data underlying that Weyl action. The prerequisites are the [adjoint representation of a Lie group](/theorems/8821), maximal tori and their normalizers, complexification of real vector spaces, simultaneous diagonalisation for commuting unitary operators, and the basic representation theory of $\mathfrak{sl}_2(\mathbb C)$. The tangent directions transverse to $T$ split into eigenspaces for the adjoint action of $T$, and the corresponding characters are the roots. From these roots we recover the reflection geometry that will control conjugacy, chambers, dominant weights, and the character formula in later chapters. ## Complexifying the Lie Algebra The adjoint representation of $T$ on the real Lie algebra $\mathfrak g$ is orthogonal, but its eigenvalues need not be real. The first problem is therefore to move to a setting where the commuting family $\operatorname{Ad}(t)$, $t\in T$, can be diagonalised simultaneously without losing the information contained in the original compact group. [definition: Complexification of a Real Lie Algebra] Let $\mathfrak g$ be a real Lie algebra. Its complexification is the complex vector space \begin{align*} \mathfrak g_{\mathbb C} := \mathfrak g \otimes_{\mathbb R} \mathbb C \end{align*} with complex-bilinear Lie bracket \begin{align*} [\ ,\ ]_{\mathfrak g_{\mathbb C}}:\mathfrak g_{\mathbb C}\times\mathfrak g_{\mathbb C}\to\mathfrak g_{\mathbb C} \end{align*} determined by \begin{align*} [X\otimes z, Y\otimes w] = [X,Y]\otimes zw \end{align*} for $X,Y\in\mathfrak g$ and $z,w\in\mathbb C$. [/definition] The real algebra $\mathfrak g$ sits inside $\mathfrak g_{\mathbb C}$ as $\mathfrak g\otimes 1$. Complexification is not a change of group; it is a linear algebra device that lets the torus action be analysed through characters. If $\mathfrak t=\operatorname{Lie}(T)$, then $\mathfrak t_{\mathbb C}$ remains an abelian subalgebra of $\mathfrak g_{\mathbb C}$. [example: Complexification of SU(2)] For the diagonal maximal torus $T=\{\operatorname{diag}(e^{i\theta},e^{-i\theta}):\theta\in\mathbb R\}$, write $E_{ij}$ for the matrix unit and set $H=E_{11}-E_{22}$, $E=E_{12}$, and $F=E_{21}$ in $\mathfrak{sl}_2(\mathbb C)$. The real Lie algebra $\mathfrak{su}(2)$ has real basis $iH$, $E-F$, and $i(E+F)$. In its complex span, \begin{align*} H=-i(iH) \end{align*} \begin{align*} E=\frac{1}{2}\big((E-F)-i\,i(E+F)\big) \end{align*} \begin{align*} F=\frac{1}{2}\big(-(E-F)-i\,i(E+F)\big), \end{align*} so $\mathfrak{su}(2)\otimes_{\mathbb R}\mathbb C$ identifies with the complex span of $H,E,F$, namely $\mathfrak{sl}_2(\mathbb C)$. Using $[E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{li}E_{kj}$, we get \begin{align*} [H,E]=[E_{11}-E_{22},E_{12}]=E_{12}-(-E_{12})=2E \end{align*} \begin{align*} [H,F]=[E_{11}-E_{22},E_{21}]=-E_{21}-E_{21}=-2F \end{align*} \begin{align*} [E,F]=[E_{12},E_{21}]=E_{11}-E_{22}=H. \end{align*} The Lie algebra of $T$ is $\mathbb R\,iH$, whose complexification is $\mathbb C H$. For $t_\theta=\operatorname{diag}(e^{i\theta},e^{-i\theta})$, conjugation gives \begin{align*} t_\theta E_{ij}t_\theta^{-1}=d_i d_j^{-1}E_{ij} \end{align*} where $d_1=e^{i\theta}$ and $d_2=e^{-i\theta}$. Hence \begin{align*} \operatorname{Ad}(t_\theta)H=H \end{align*} \begin{align*} \operatorname{Ad}(t_\theta)E=e^{i\theta}e^{i\theta}E=e^{2i\theta}E \end{align*} \begin{align*} \operatorname{Ad}(t_\theta)F=e^{-i\theta}e^{-i\theta}F=e^{-2i\theta}F. \end{align*} Thus $\mathbb C H$ is the zero-weight direction coming from the torus, while $\mathbb C E$ and $\mathbb C F$ are the two opposite non-zero eigendirections for the adjoint action of $T$. [/example] This example foreshadows the general picture: the torus directions remain fixed, and all other directions occur in opposite pairs. To make this precise, we need a language for simultaneous eigenspaces of a torus representation. [definition: Weight Space for the Adjoint Action] Let $T$ be a torus acting complex-linearly on a finite-dimensional complex vector space $V$. For a character $\lambda:T\to S^1$, the $\lambda$-weight space is \begin{align*} V_\lambda := \{v\in V: t\cdot v=\lambda(t)v\text{ for all }t\in T\}. \end{align*} [/definition] The definition gives candidate eigenspaces, but it does not by itself say that they exhaust $V$. For the root construction to be meaningful, the adjoint representation cannot have hidden directions left over after all character eigenspaces have been collected. The obstruction is possible non-diagonal behaviour: simultaneous eigenspaces might fail to span unless compactness and finite-dimensionality force complete reducibility. The needed input is that finite-dimensional complex representations of a compact torus decompose completely into their weight spaces. [quotetheorem:9726] [citeproof:9726] Applied to $V=\mathfrak g_{\mathbb C}$, this theorem separates the directions fixed by $T$ from the directions on which $T$ rotates by a non-zero character. Compactness is doing real work here: without a unitary structure, commuting linear maps need not be diagonalisable, as a single nonzero nilpotent Jordan block already shows. Finite-dimensionality is also essential: on infinite-dimensional spaces a commuting unitary family can have continuous spectrum, such as the rotation action of $S^1$ on $L^2(S^1)$ before restricting to finite Fourier sums, so an algebraic direct sum of weight spaces need not exhaust the representation. The complex scalar field is what lets rotations split into eigenspaces; over $\mathbb R$, the same action may remain as two-dimensional rotation planes rather than one-dimensional weight spaces. The theorem does not identify which characters occur, nor does it say that their weight spaces are one-dimensional. It only supplies the decomposition needed to ask the next question: which non-zero weights occur in the adjoint representation? The next section gives those non-zero weights their geometric name. ## Roots and Root Spaces The maximal torus is supposed to encode the group up to finite symmetry, but the Lie algebra still contains directions not tangent to the torus. The question is how those transverse directions are organised by the adjoint action of $T$, and which part of the resulting data is intrinsic rather than a choice of coordinates. [definition: Root and Root Space] Let $G$ be a compact connected Lie group with maximal torus $T$, and let $\mathfrak g_{\mathbb C}$ be the complexified Lie algebra. A root of $(G,T)$ is a non-zero character $\alpha:T\to S^1$ such that \begin{align*} \mathfrak g_\alpha := \{X\in\mathfrak g_{\mathbb C}:\operatorname{Ad}(t)X=\alpha(t)X\text{ for all }t\in T\} \end{align*} is non-zero. The space $\mathfrak g_\alpha$ is the root space associated to $\alpha$. [/definition] From now on we use the same symbol $\alpha$ for both the group character $\alpha:T\to S^1$ and its differentiated weight $d\alpha:\mathfrak t_{\mathbb C}\to\mathbb C$; addition such as $\alpha+\beta$, evaluation such as $\beta(H_\alpha)$, and kernels such as $\ker\alpha$ refer to these differentiated weights. Equivalently, the roots will often be regarded as elements of the real vector space spanned by the differentiated weights inside $i\mathfrak t^*$. This convention is essential because characters multiply on $T$, while root systems are written additively in the dual of the Lie algebra. The zero-weight space is not called a root space in this convention. A naive decomposition could have a larger fixed part if $T$ were not maximal: for example, a smaller torus inside a larger torus fixes extra toral directions. There is a second, subtler issue: a compact connected group may have a central torus, and roots vanish on those central directions. Thus roots need not span all of $\mathfrak t^*$ for the original group. The semisimple root-space theorem below should therefore be read on the derived, semisimple part of the Lie algebra, or equivalently after separating off the central torus. In that reduced setting, roots do serve as a substitute for the transverse adjoint representation, and the bracket records how the corresponding weights add. [quotetheorem:4685] [citeproof:4685] For the original compact connected group, the theorem supplies the noncentral part of the dictionary. The central torus contributes fixed toral directions on which every root is zero, while the semisimple summand supplies the finite root system $\Phi$ and its root spaces. Maximality of $T$ is still essential: it ensures that the fixed part of the adjoint action is the toral direction rather than an artefact of choosing too small a torus. At this point in the course we have not yet proved the later rank-one consequences, including one-dimensionality of root spaces. The bracket relation says that the semisimple Lie algebra remembers addition among roots, so the set of roots cannot be arbitrary, and it is this additive constraint that makes root strings possible. [example: The Root Pair for SU(2)] For $G=SU(2)$, take the diagonal maximal torus \begin{align*} T=\{\operatorname{diag}(e^{i\theta},e^{-i\theta}):\theta\in\mathbb R\}. \end{align*} Write \begin{align*} E=E_{12},\qquad F=E_{21},\qquad H=E_{11}-E_{22}. \end{align*} For $t_\theta=\operatorname{diag}(e^{i\theta},e^{-i\theta})$, the inverse is \begin{align*} t_\theta^{-1}=\operatorname{diag}(e^{-i\theta},e^{i\theta}). \end{align*} Since left multiplication by a diagonal matrix multiplies the $i$th row by its $i$th diagonal entry, and right multiplication by a diagonal matrix multiplies the $j$th column by its $j$th diagonal entry, we get \begin{align*} t_\theta E_{12}t_\theta^{-1}=e^{i\theta}e^{i\theta}E_{12}=e^{2i\theta}E. \end{align*} Similarly, \begin{align*} t_\theta E_{21}t_\theta^{-1}=e^{-i\theta}e^{-i\theta}E_{21}=e^{-2i\theta}F. \end{align*} Also, \begin{align*} t_\theta E_{11}t_\theta^{-1}=e^{i\theta}e^{-i\theta}E_{11}=E_{11} \end{align*} and \begin{align*} t_\theta E_{22}t_\theta^{-1}=e^{-i\theta}e^{i\theta}E_{22}=E_{22}, \end{align*} so \begin{align*} \operatorname{Ad}(t_\theta)H=H. \end{align*} Thus $\mathbb C H$ is the zero-weight space, $\mathbb C E$ has character $\alpha(t_\theta)=e^{2i\theta}$, and $\mathbb C F$ has character $-\alpha$ in additive notation, meaning $(-\alpha)(t_\theta)=e^{-2i\theta}$. Therefore the non-zero roots are exactly the pair $\{\alpha,-\alpha\}$, with root spaces \begin{align*} \mathfrak g_\alpha=\mathbb C E \end{align*} and \begin{align*} \mathfrak g_{-\alpha}=\mathbb C F. \end{align*} This is the rank-one pattern: the torus direction is fixed, and the two transverse eigendirections occur with opposite characters. [/example] The $SU(2)$ calculation is the local model inside every compact Lie group. Each root determines a copy of the rank-one structure generated by $\mathfrak g_\alpha$, $\mathfrak g_{-\alpha}$, and their bracket in $\mathfrak t_{\mathbb C}$. ## Coroots and the Rank-One Subalgebra Roots are characters, hence they naturally live in a lattice dual to the torus. A root alone tells us the hyperplane it vanishes on, but it does not by itself specify the correct normal vector or the scale of the reflection. Choosing an arbitrary normal would give non-integral reflection formulae and would not match the rank-one $SU(2)$ subgroups inside $G$. To turn a root into a reflection, we need the distinguished normal direction supplied by the Lie bracket of the opposite root spaces. This direction is the coroot. [definition: Coroot] Let $G$ be compact and connected, let $T$ be a maximal torus, and let $\alpha\in\Phi$. Write $d\alpha:\mathfrak t_{\mathbb C}\to\mathbb C$ for the differentiated root. The coroot $H_\alpha\in\mathfrak t_{\mathbb C}$ is the element determined by the rank-one subalgebra generated by $\mathfrak g_\alpha$ and $\mathfrak g_{-\alpha}$, normalised so that \begin{align*} d\alpha(H_\alpha)=2. \end{align*} [/definition] The compact real form supplies an invariant inner product on $\mathfrak g$, so this normalisation can also be expressed by pairing $\mathfrak t$ with its dual. The problem is that a root by itself only gives a functional; it does not automatically produce the opposite root vector or the distinguished element of $\mathfrak t_{\mathbb C}$ on which the root takes the value $2$. To make the coroot canonical rather than a chosen normal vector, the opposite root spaces must assemble into a rank-one subalgebra. [quotetheorem:4689] [citeproof:4689] This rank-one subalgebra is the bridge from compact Lie groups to the representation theory of $\mathfrak{sl}_2(\mathbb C)$. The compactness hypothesis enters through the conjugation pairing opposite root spaces; without such a pairing, the opposite root direction and the normalisation need not be canonical. A concrete failure occurs in a solvable complex Lie algebra with basis $H,E$ and bracket $[H,E]=E$: there is a positive weight space for the action of $\mathbb C H$, but no opposite root space and hence no element $F$ with $[E,F]=H$. The theorem does not yet classify the whole Lie algebra: it only identifies a rank-one subalgebra attached to a single root. Its importance is that every other root space can now be tested against this copy of $\mathfrak{sl}_2(\mathbb C)$, which is the source of integrality and root strings. This is the same mechanism behind angular momentum decompositions in mathematical physics: a large symmetry representation is understood by restricting it to many embedded $SU(2)$ subgroups. [example: Coroot in SL(2)] In $\mathfrak{sl}_2(\mathbb C)$, take \begin{align*} E=E_{12},\qquad F=E_{21},\qquad H=E_{11}-E_{22}. \end{align*} Using $E_{ij}E_{kl}=\delta_{jk}E_{il}$, we compute \begin{align*} EF=E_{12}E_{21}=E_{11} \end{align*} and \begin{align*} FE=E_{21}E_{12}=E_{22}. \end{align*} Hence \begin{align*} [E,F]=EF-FE=E_{11}-E_{22}=H. \end{align*} Similarly, \begin{align*} HE=(E_{11}-E_{22})E_{12}=E_{12}-0=E \end{align*} and \begin{align*} EH=E_{12}(E_{11}-E_{22})=0-E_{12}=-E, \end{align*} so \begin{align*} [H,E]=HE-EH=E-(-E)=2E. \end{align*} Also, \begin{align*} HF=(E_{11}-E_{22})E_{21}=0-E_{21}=-F \end{align*} and \begin{align*} FH=E_{21}(E_{11}-E_{22})=E_{21}-0=F, \end{align*} so \begin{align*} [H,F]=HF-FH=-F-F=-2F. \end{align*} Thus the positive root $\alpha$ is the weight of the line $\mathbb C E$, because $[H,E]=2E$ means $\alpha(H)=2$. The coroot is therefore $H_\alpha=H$, since it is normalised precisely by the condition $\alpha(H_\alpha)=2$. [/example] The same computation occurs inside every root direction, but in higher rank other root spaces can sit in strings under this embedded $\mathfrak{sl}_2$. ## Root Strings and Integrality Once a root $\alpha$ has supplied an $\mathfrak{sl}_2$ subalgebra, every other root space becomes part of a finite-dimensional $\mathfrak{sl}_2$ representation. A naive approach would only record whether $\beta+\alpha$ is a root, but that loses information: in type $B_2$, a root may continue for different distances in the two directions along a fixed line. The problem is to understand exactly how far one can move from a root $\beta$ by repeatedly adding or subtracting $\alpha$, and whether gaps can occur along the way. [definition: Root String] Let $\alpha,\beta\in\Phi$ with $\alpha\ne\pm\beta$. The $\alpha$-string through $\beta$ is the set of roots of the form \begin{align*} \beta+k\alpha, \end{align*} where $k\in\mathbb Z$. [/definition] The definition names the set, but it does not explain its shape. The theorem below is needed because later arguments use the fact that root strings have no gaps and have integer endpoints measured by the coroot. [quotetheorem:4693] [citeproof:4693] The theorem is the first point where the discreteness of root systems becomes visible. The exclusion $\alpha\ne\pm\beta$ avoids the degenerate string through the root defining the embedded $\mathfrak{sl}_2$ itself; in those cases the rank-one subalgebra has to be treated separately. Finite-dimensionality is what prevents an infinite ladder of weights: an infinite-dimensional highest-weight module for $\mathfrak{sl}_2(\mathbb C)$ can have weights continuing indefinitely in one direction, so there would be no finite pair $p,q$. The $\mathfrak{sl}_2$-representation hypothesis is also essential; for a mere finite set of weights stable under no raising and lowering operators, the subset $\{\beta,\beta+2\alpha\}$ has a gap at $\beta+\alpha$ and violates the string conclusion. The result says that strings have no gaps, but it does not determine the actual values of $p$ and $q$ without knowing which neighbouring roots occur. Inner products among roots are nevertheless forced into a small set of possibilities because the numbers $\beta(H_\alpha)$ are integers, and this is exactly the input needed for the Cartan integers. [quotetheorem:4691] [citeproof:4691] These integers are the entries of the Cartan matrix once a basis of simple roots is chosen. The invariant inner product is needed for the displayed angle formula; the integrality statement itself comes from the $\mathfrak{sl}_2$ action and does not depend on first drawing a Euclidean diagram. Outside the root-system setting, the quotient $2(\beta,\alpha)/(\alpha,\alpha)$ need not be integral: in $\mathbb R^2$ with the standard inner product, taking $\alpha=(1,0)$ and $\beta=(1/3,1)$ gives $2/3$. Such vectors do not arise as compact Lie roots because there is no associated finite $\mathfrak{sl}_2$ string forcing the evaluation on $H_\alpha$ to be an integer. The theorem does not say that arbitrary integer values can occur, because finiteness of the root string imposes additional restrictions. Before choosing a basis, the Cartan integers already determine the possible angles and length ratios between pairs of roots, which is why only a small list of rank-two diagrams can appear. [example: The $A_2$ Root Diagram for $SU(3)$] For $G=SU(3)$, take the diagonal maximal torus \begin{align*} T=\{\operatorname{diag}(z_1,z_2,z_3):z_i\in S^1,\ z_1z_2z_3=1\}. \end{align*} Its complexified Lie algebra has diagonal part \begin{align*} \mathfrak t_{\mathbb C}=\{\operatorname{diag}(x_1,x_2,x_3):x_1+x_2+x_3=0\}, \end{align*} and we write $\varepsilon_i(\operatorname{diag}(x_1,x_2,x_3))=x_i$. For the matrix unit $E_{ij}$ with $i\ne j$, left multiplication by $t=\operatorname{diag}(z_1,z_2,z_3)$ multiplies row $i$ by $z_i$, while right multiplication by $t^{-1}$ multiplies column $j$ by $z_j^{-1}$. Hence \begin{align*} tE_{ij}t^{-1}=z_i z_j^{-1}E_{ij}. \end{align*} Thus the line $\mathbb C E_{ij}$ has character $t\mapsto z_i z_j^{-1}$, whose differentiated weight is $\varepsilon_i-\varepsilon_j$. The off-diagonal directions in $\mathfrak{sl}_3(\mathbb C)$ are \begin{align*} \mathbb C E_{12},\mathbb C E_{21},\mathbb C E_{23},\mathbb C E_{32},\mathbb C E_{13},\mathbb C E_{31}. \end{align*} Their roots are respectively \begin{align*} \varepsilon_1-\varepsilon_2,\ \varepsilon_2-\varepsilon_1,\ \varepsilon_2-\varepsilon_3,\ \varepsilon_3-\varepsilon_2,\ \varepsilon_1-\varepsilon_3,\ \varepsilon_3-\varepsilon_1. \end{align*} Therefore \begin{align*} \Phi=\{\pm(\varepsilon_1-\varepsilon_2),\pm(\varepsilon_2-\varepsilon_3),\pm(\varepsilon_1-\varepsilon_3)\}. \end{align*} Inside the plane $\varepsilon_1+\varepsilon_2+\varepsilon_3=0$, using the inner product inherited from $\mathbb R^3$, each root $\varepsilon_i-\varepsilon_j$ has squared length \begin{align*} (\varepsilon_i-\varepsilon_j,\varepsilon_i-\varepsilon_j)=1+1=2. \end{align*} If $\alpha_1=\varepsilon_1-\varepsilon_2$ and $\alpha_1+\alpha_2=\varepsilon_1-\varepsilon_3$, then \begin{align*} (\alpha_1,\alpha_1+\alpha_2)=(\varepsilon_1-\varepsilon_2,\varepsilon_1-\varepsilon_3)=1. \end{align*} Since both roots have length $\sqrt2$, the cosine of the angle between them is $1/2$, so adjacent roots meet at angle $60^\circ$. Thus the six roots have equal length and occur at successive angles of $60^\circ$, forming a regular hexagon. [/example] The $A_2$ picture is the smallest example where adding roots matters. It also shows why reflections should appear: the hexagon is generated by reflecting across hyperplanes perpendicular to simple roots. ## Reflections from Roots A root gives a hyperplane $\ker\alpha$ in the real vector space associated to $\mathfrak t$, and the coroot gives the normal direction needed to reflect across that hyperplane. The main question is whether these geometric reflections are external symmetries of the diagram or actual Weyl group elements coming from the compact Lie group. [definition: Root Reflection] Let $\alpha\in\Phi$, and let $E=\operatorname{span}_{\mathbb R}(\Phi)$. The root reflection associated to $\alpha$ is the map $s_\alpha:E\to E$ defined by \begin{align*} s_\alpha(\beta)=\beta-\beta(H_\alpha)\alpha \end{align*} for $\beta\in E$, extending linearly from the roots. [/definition] Because $\beta(H_\alpha)$ is an integer, this reflection preserves the root lattice. Preserving the lattice is only an abstract combinatorial property; it does not yet show that the reflection is realized by conjugation inside the compact group. The obstruction is that a symmetry of the root diagram could be external to $G$, while the Weyl group only contains symmetries coming from elements of $N_G(T)$. The next point is therefore to connect the rank-one subgroup attached to $\alpha$ with an actual normalizer element whose action on roots is exactly the reflection formula. [quotetheorem:9727] [citeproof:9727] Thus the finite Weyl group is not an additional structure placed on the root system; the reflections forced by roots already occur inside it. The hypothesis that the rank-one subgroup lies in the compact group is what turns an abstract reflection formula into an element of $N_G(T)/T$. The theorem does not say that these reflections already generate all of $W$; that requires a chamber argument after simple roots have been chosen. This is the mechanism by which local $SU(2)$ subgroups generate global symmetries, and it is also the point where the geometry of the root diagram becomes representation-theoretic. [example: The Reflection in SU(2)] Take the element $n=E_{21}-E_{12}\in SU(2)$. Since $n^2=-I$, we have $n^{-1}=-n=E_{12}-E_{21}$. For \begin{align*} t_\theta=e^{i\theta}E_{11}+e^{-i\theta}E_{22}, \end{align*} the matrix-unit rule $E_{ij}E_{kl}=\delta_{jk}E_{il}$ gives \begin{align*} nt_\theta=e^{i\theta}E_{21}-e^{-i\theta}E_{12}. \end{align*} Multiplying by $n^{-1}$ on the right, \begin{align*} nt_\theta n^{-1}=(e^{i\theta}E_{21}-e^{-i\theta}E_{12})(E_{12}-E_{21})=e^{i\theta}E_{22}+e^{-i\theta}E_{11}. \end{align*} Thus \begin{align*} nt_\theta n^{-1}=\operatorname{diag}(e^{-i\theta},e^{i\theta})=t_{-\theta}, \end{align*} so $n$ normalises the diagonal torus and represents the non-identity Weyl element. If $\alpha(t_\theta)=e^{2i\theta}$ is the root on the line $\mathbb C E_{12}$, then the induced action on characters satisfies \begin{align*} (n\cdot\alpha)(t_\theta)=\alpha(n^{-1}t_\theta n)=\alpha(t_{-\theta})=e^{-2i\theta}=(-\alpha)(t_\theta). \end{align*} The same computation sends $-\alpha$ back to $\alpha$, so the Weyl action on $\{\alpha,-\alpha\}$ is exactly the rank-one reflection interchanging the two roots. [/example] In rank one this is the whole Weyl group. In higher rank the content is that many such rank-one reflections coexist and satisfy the axioms of a finite crystallographic reflection system. ## The Root System Axioms We have constructed roots from a compact connected Lie group, but the next stages of the course will often reason only with the resulting finite subset of a Euclidean vector space. The task is to isolate the properties of $\Phi$ that are guaranteed by the Lie-theoretic construction. [definition: Root System] Let $E$ be a finite-dimensional real inner product space. A finite subset $\Phi\subset E\setminus\{0\}$ is a root system if it spans $E$, if $\Phi\cap\mathbb R\alpha=\{\alpha,-\alpha\}$ for every $\alpha\in\Phi$, if $s_\alpha(\Phi)=\Phi$ for every $\alpha\in\Phi$, and if \begin{align*} \frac{2(\beta,\alpha)}{(\alpha,\alpha)}\in\mathbb Z \end{align*} for all $\alpha,\beta\in\Phi$. [/definition] For compact connected groups, the relevant Euclidean space is the real span of the roots inside $i\mathfrak t^*$, equipped with the inner product induced by an invariant inner product on $\mathfrak g$. The following theorem is needed to justify replacing Lie-theoretic arguments by root-system arguments in the classification part of the course. [quotetheorem:9728] [citeproof:9728] This theorem allows us to use the language of abstract root systems without forgetting where the objects came from. Compactness is used to obtain invariant inner products and rank-one compact subgroups; connectedness is used so that maximal tori and their normalisers control the group rather than only the identity component. Working in the real span of the roots deliberately discards central torus directions, so the theorem describes the semisimple reflection data and not the whole torus of $G$. The same finite reflection diagrams also appear in the classification of complex semisimple Lie algebras, in the geometry of symmetric spaces, and in the weight diagrams used in quantum mechanics to organise particles by symmetry. Later classification results proceed by classifying such root systems and then reconstructing compact groups up to finite central data, with the missing central information handled separately. [example: A $B_2$ Root System from $SO(5)$ or $Sp(2)$] In $\mathbb R^2$ with standard orthonormal basis $e_1,e_2$, consider \begin{align*} \Phi=\{\pm e_1,\pm e_2,\pm(e_1+e_2),\pm(e_1-e_2)\}. \end{align*} The short roots are $\pm e_1,\pm e_2$, since \begin{align*} (e_1,e_1)=1 \end{align*} and the same computation gives $(e_2,e_2)=1$. The long roots are $\pm(e_1+e_2),\pm(e_1-e_2)$, since \begin{align*} (e_1+e_2,e_1+e_2)=(e_1,e_1)+2(e_1,e_2)+(e_2,e_2)=1+0+1=2 \end{align*} and \begin{align*} (e_1-e_2,e_1-e_2)=(e_1,e_1)-2(e_1,e_2)+(e_2,e_2)=1-0+1=2. \end{align*} For the short root $e_1$, the reflection formula gives \begin{align*} s_{e_1}(x_1e_1+x_2e_2)=x_1e_1+x_2e_2-2x_1e_1=-x_1e_1+x_2e_2. \end{align*} Similarly, \begin{align*} s_{e_2}(x_1e_1+x_2e_2)=x_1e_1-x_2e_2. \end{align*} For the long root $e_1-e_2$, we have $(e_1-e_2,e_1-e_2)=2$ and \begin{align*} (x_1e_1+x_2e_2,e_1-e_2)=x_1-x_2, \end{align*} so \begin{align*} s_{e_1-e_2}(x_1e_1+x_2e_2)=x_1e_1+x_2e_2-(x_1-x_2)(e_1-e_2)=x_2e_1+x_1e_2. \end{align*} Thus the root reflections include both independent sign changes and the coordinate swap. These generate all signed permutations \begin{align*} (x_1,x_2)\mapsto(\pm x_1,\pm x_2) \end{align*} and \begin{align*} (x_1,x_2)\mapsto(\pm x_2,\pm x_1), \end{align*} which are exactly the eight symmetries of a square. This is the $B_2$ root system: it occurs for $SO(5)$, while the dual picture occurs for $Sp(2)$ depending on whether one draws roots or coroots. [/example] The contrast between $A_2$ and $B_2$ is the first sign that root systems remember length data, not just angles. This length data affects coroots and therefore later affects weights and characters. ## Simple Roots and Generation of the Weyl Group The full root system is symmetric around the origin, which is too redundant for many calculations. If we try to list all roots at once, even the $A_2$ example repeats the same information in positive and negative directions, and there is no preferred way to write a highest weight or a character. To compute with roots we choose a notion of positivity, then extract the indecomposable positive roots. The problem is to make this choice in a way that divides the vector space into chambers and identifies a minimal set of generating reflections. [definition: Positive and Simple Roots] Let $\Phi$ be a root system in a real inner product space $E$. A choice of positive roots is a subset $\Phi^+\subset\Phi$ obtained from some vector $h\in E$ with $(\alpha,h)\ne 0$ for all $\alpha\in\Phi$, by declaring \begin{align*} \Phi^+ := \{\alpha\in\Phi:(\alpha,h)>0\}. \end{align*} A positive root $\alpha\in\Phi^+$ is simple if it cannot be written as $\alpha=\beta+\gamma$ with $\beta,\gamma\in\Phi^+$. [/definition] This hyperplane construction ensures that if $\alpha,\beta,\alpha+\beta\in\Phi$ and $\alpha,\beta$ are positive, then $\alpha+\beta$ is positive as well. The first issue is whether the indecomposable positive roots really behave like a basis rather than just a convenient list: later calculations require every positive root to be built from them with nonnegative integer coefficients, and they must be independent enough for those coefficients to be meaningful. [quotetheorem:4700] [citeproof:4700] The theorem replaces a symmetric root diagram by a directed combinatorial basis. Its content is stronger than saying that the simple roots span the ambient vector space: it says that every positive root has a unique expression using nonnegative integer coefficients of the simple roots. This uniqueness is what makes later notions such as height, dominance, and highest weight comparisons well defined. Finiteness of $\Phi$ is essential in the descent argument; without it, repeated subtraction need not terminate. The positivity condition is essential as well: if one takes the infinite set $\{n\alpha:n\in\mathbb Z\setminus\{0\}\}$ on a line and declares the positive roots to be $n\alpha$ for $n>0$, then no finite simple basis can generate all positive roots by the same root-system argument. Even for finite symmetric sets, choosing one vector from each pair without a separating hyperplane can break closure under addition, so an indecomposable list need not control all positive roots. The theorem does not make the choice of $\Delta$ canonical, since a different chamber gives a different simple system, but it does ensure that once the chamber is fixed, all positive-root bookkeeping is reduced to the simple roots. Once the positive roots have been reduced to simple ones, a second problem remains. The Weyl group was defined by reflections coming from the group, and it is not automatic that the reflections attached only to the simple roots recover all of its chamber-moving symmetries. The next statement isolates the chamber geometry that makes the simple system control the full reflection structure. [quotetheorem:4699] [citeproof:4699] This chamber statement explains why the simple system is not merely a list of generators for a monoid of positive roots. It also marks out the walls of a fundamental region for the reflection action. In the $A_2$ picture, the two simple roots determine the two bounding walls of one chamber, and reflecting across those walls moves to adjacent chambers. The theorem is limited to the chosen positive system: changing the chamber changes which roots are simple, although the resulting chambers remain part of the same Weyl-group geometry. What remains is to convert this geometric control of chambers into an algebraic generating statement for the Weyl group itself. [quotetheorem:4705] [citeproof:4705] This generation result is the endpoint of the chapter. The chamber argument uses the positivity choice: without a fundamental chamber there is no length-reduction procedure that singles out the simple reflections. The statement concerns the action on the root span, or equivalently on the quotient of $\mathfrak t$ by the central directions killed by every root. For a compact group with a nonzero central torus, such as $U(n)$, the scalar circle contributes a torus direction on which all roots vanish, while the Weyl group acts faithfully on the semisimple quotient direction coming from diagonal entries modulo the scalar line. Starting from the adjoint action of a maximal torus, we obtained roots, root strings, reflections, and finally a finite Coxeter-type generating set for the Weyl group, setting up the later construction of dominant weights and characters. [example: Simple Reflections in A2] For $SU(3)$, take the positive roots \begin{align*} \alpha_1=\varepsilon_1-\varepsilon_2,\qquad \alpha_2=\varepsilon_2-\varepsilon_3,\qquad \alpha_1+\alpha_2=\varepsilon_1-\varepsilon_3. \end{align*} With the inner product inherited from $\mathbb R^3$, we have \begin{align*} (\alpha_1,\alpha_1)=(\varepsilon_1-\varepsilon_2,\varepsilon_1-\varepsilon_2)=1+1=2. \end{align*} Also \begin{align*} (\alpha_2,\alpha_1)=(\varepsilon_2-\varepsilon_3,\varepsilon_1-\varepsilon_2)=-1. \end{align*} Therefore the reflection formula gives \begin{align*} s_{\alpha_1}(\alpha_2)=\alpha_2-\frac{2(\alpha_2,\alpha_1)}{(\alpha_1,\alpha_1)}\alpha_1=\alpha_2-\frac{2(-1)}{2}\alpha_1=\alpha_1+\alpha_2. \end{align*} Similarly, \begin{align*} s_{\alpha_1}(\alpha_1)=\alpha_1-\frac{2(\alpha_1,\alpha_1)}{(\alpha_1,\alpha_1)}\alpha_1=-\alpha_1. \end{align*} Thus $s_{\alpha_1}$ swaps the first two weights, so it is the transposition $(12)$ on the diagonal entries. For $\alpha_2$, the same calculation gives \begin{align*} s_{\alpha_2}(\alpha_1)=\alpha_1-\frac{2(\alpha_1,\alpha_2)}{(\alpha_2,\alpha_2)}\alpha_2=\alpha_1-\frac{2(-1)}{2}\alpha_2=\alpha_1+\alpha_2 \end{align*} and \begin{align*} s_{\alpha_2}(\alpha_2)=-\alpha_2. \end{align*} So $s_{\alpha_2}$ swaps the second and third weights, giving the transposition $(23)$. The transpositions $(12)$ and $(23)$ generate \begin{align*} S_3=\{1,(12),(23),(123),(132),(13)\}, \end{align*} so the two simple reflections generate the Weyl group. Concretely, this is the Weyl action by permutation matrices, which permute the three diagonal entries of the maximal torus. [/example] The adjoint representation reveals the root system hidden in the torus action, and the simple reflections of the Weyl group become visible as permutations of those roots. The next chapter packages this structure into lattices of characters, cocharacters, roots, and coroots, which record both the continuous and discrete data of the group. # 5. Lattices, Weights, and Root Data This chapter turns the conjugacy theory of maximal tori into a combinatorial invariant. Once a maximal torus $T \subset G$ has been chosen, the continuous part of the representation theory is recorded by lattices of characters, cocharacters, roots, and coroots. The new question is how much of the compact connected Lie group can be recovered from these integral structures, and how finite central quotients alter them without changing the local Lie algebra. ## Characters and Cocharacters of a Maximal Torus Chapters 2 and 3 showed that every element of a compact connected Lie group is conjugate into a maximal torus, and that the remaining ambiguity is measured by the Weyl group. To use this in representation theory, we need a way to measure maps from the torus to the circle by integer data rather than by coordinates depending on a chosen parametrisation. [definition: Character Lattice] Let $T$ be a compact torus. The character lattice of $T$ is \begin{align*} X^*(T) := \operatorname{Hom}_{\mathrm{LieGrp}}(T, U(1)). \end{align*} [/definition] Pointwise multiplication makes $X^*(T)$ an abelian group. A character is a one-dimensional complex representation of $T$. Since $T$ is abelian, irreducible complex representations of $T$ are exactly these characters, so $X^*(T)$ is the weight bookkeeping device for restrictions of representations from $G$ to $T$. We also need the dual integer object: instead of functions out of $T$, we must record the circle subgroups inside $T$, because roots will later be tested against those circles. [definition: Cocharacter Lattice] Let $T$ be a compact torus. The cocharacter lattice of $T$ is \begin{align*} X_*(T) := \operatorname{Hom}_{\mathrm{LieGrp}}(U(1), T). \end{align*} [/definition] Pointwise multiplication in $T$ makes $X_*(T)$ an abelian group. Characters test points of the torus, while cocharacters draw circles inside it. To compare these two kinds of data, compose a character with a cocharacter; the result is a self-map of $U(1)$, and such maps are classified by an integer degree. This integer is the basic pairing used throughout the chapter. [definition: Character Cocharacter Pairing] The character-cocharacter pairing is the map \begin{align*} \langle \cdot,\cdot\rangle:X^*(T)\times X_*(T)\to\mathbb Z \end{align*} defined as follows. For $\lambda \in X^*(T)$ and $\mu \in X_*(T)$, \begin{align*} \langle \lambda, \mu \rangle := n, \end{align*} where $(\lambda \circ \mu)(z)=z^n$ for all $z \in U(1)$. [/definition] This pairing is perfect for compact tori: after choosing an isomorphism $T \cong (U(1))^r$, both lattices identify with $\mathbb Z^r$, and the pairing is the standard dot product. The intrinsic notation matters because later constructions, especially roots and weights, must not depend on a chosen coordinate system. A concrete model fixes the normalisation before we move to the exponential map. [example: Standard Torus] Let $T=(U(1))^r$. For $m=(m_1,\dots,m_r)\in\mathbb Z^r$, define $\lambda_m:T\to U(1)$ by \begin{align*} \lambda_m(z_1,\dots,z_r)=z_1^{m_1}\cdots z_r^{m_r}. \end{align*} This is a character because multiplication in $T$ is coordinatewise. Conversely, if $\lambda:T\to U(1)$ is any character, restrict it to the $i$th coordinate circle: \begin{align*} \lambda_i(z):=\lambda(1,\dots,1,z,1,\dots,1). \end{align*} Each character of $U(1)$ has the form $z\mapsto z^{m_i}$ for a unique $m_i\in\mathbb Z$. Since \begin{align*} (z_1,\dots,z_r)=(z_1,1,\dots,1)(1,z_2,1,\dots,1)\cdots(1,\dots,1,z_r), \end{align*} the homomorphism property gives \begin{align*} \lambda(z_1,\dots,z_r)=z_1^{m_1}z_2^{m_2}\cdots z_r^{m_r}. \end{align*} Thus $X^*(T)\cong\mathbb Z^r$ by $m\mapsto\lambda_m$. Similarly, for $n=(n_1,\dots,n_r)\in\mathbb Z^r$, define $\mu_n:U(1)\to T$ by \begin{align*} \mu_n(z)=(z^{n_1},\dots,z^{n_r}). \end{align*} Every cocharacter has this form, because composing a cocharacter $\mu:U(1)\to T$ with the $i$th projection $T\to U(1)$ gives a character $z\mapsto z^{n_i}$ of $U(1)$. Now \begin{align*} (\lambda_m\circ\mu_n)(z)=\lambda_m(z^{n_1},\dots,z^{n_r}). \end{align*} Substituting the definition of $\lambda_m$ gives \begin{align*} \lambda_m(z^{n_1},\dots,z^{n_r})=(z^{n_1})^{m_1}\cdots(z^{n_r})^{m_r}. \end{align*} Using $(z^a)^b=z^{ab}$ and multiplying powers of the same element, \begin{align*} (z^{n_1})^{m_1}\cdots(z^{n_r})^{m_r}=z^{m_1n_1+\cdots+m_rn_r}. \end{align*} Therefore \begin{align*} \langle\lambda_m,\mu_n\rangle=m_1n_1+\cdots+m_rn_r. \end{align*} In these coordinates, the character and cocharacter lattices are dual copies of $\mathbb Z^r$, and the pairing is the standard dot product. [/example] The standard torus example shows that characters are controlled by integer exponents. For an arbitrary compact torus, the same integrality must be expressed without coordinates, and the exponential map supplies the missing object: its kernel is the period lattice. The next result explains exactly which linear functionals on the Lie algebra exponentiate to genuine characters. [quotetheorem:9729] [citeproof:9729] Thus the character lattice is the integral dual of the exponential kernel. The hypotheses are doing real work. Compactness gives a torus of the form $\mathfrak t/\Lambda_T$ with $\Lambda_T$ a full lattice, so periods are discrete and characters have integer winding numbers. For the non-compact additive group $\mathbb R$, the homomorphism $x\mapsto e^{2\pi iax}$ is a character for every $a\in\mathbb R$, because the exponential kernel is absent; there is no integrality condition to impose. Conversely, if a formula on $\mathfrak t$ is not invariant under the exponential kernel, it defines a character on the covering vector group but does not descend to $T$: on $U(1)=\mathbb R/\mathbb Z$, the formula $x\mapsto e^{2\pi i x/2}$ changes sign when $x$ is replaced by $x+1$ and therefore is not a character of $U(1)$. The theorem also does not say that every real linear functional has a compact-torus character attached to it. It singles out the integral dual lattice inside $\mathfrak t^*$, and changing the global torus by changing its period lattice changes that integral lattice. This is the first place where two groups with the same Lie algebra may differ: their exponential kernels, and hence their allowed characters, need not be the same. We now apply the same lattice language to representations of a compact group by decomposing them under the maximal torus. ## Weight, Root, and Coroot Lattices Representations of $G$ restrict to representations of $T$, and those restrictions split into characters. The problem is to isolate which torus characters arise from the adjoint representation, which arise from arbitrary representations, and which integral lattice records the global form of $G$. [definition: Weight Space] Let $G$ be a compact connected Lie group with maximal torus $T$, and let $V$ be a finite-dimensional complex representation of $G$. For $\lambda\in X^*(T)$, the $\lambda$-weight space of $V$ is \begin{align*} V_\lambda := \{v\in V : t\cdot v=\lambda(t)v\text{ for all }t\in T\}. \end{align*} [/definition] The weights of $V$ are the characters $\lambda$ such that $V_\lambda\ne 0$. Since $T$ is compact and abelian, $V$ decomposes as a direct sum of its weight spaces. This gives a vocabulary for individual representations, but we also need the ambient lattice in which all possible weights for the chosen global group live. That ambient lattice is the character lattice of the maximal torus. [definition: Weight Lattice of a Compact Group] Let $G$ be a compact connected Lie group with maximal torus $T$. The weight lattice of $G$ relative to $T$ is the subgroup \begin{align*} P_G := X^*(T). \end{align*} [/definition] For the representation theory of a fixed global group $G$, the full character lattice of its maximal torus is the relevant weight lattice. A smaller and more rigid set of weights comes from the adjoint representation of $G$ on its own Lie algebra. We now need to define these special non-zero adjoint weights because they encode the non-abelian directions transverse to the torus. [definition: Root System] Let $G$ be a compact connected Lie group with maximal torus $T$ and complexified Lie algebra $\mathfrak g_{\mathbb C}$. A root is a non-zero character $\alpha\in X^*(T)$ such that \begin{align*} \mathfrak g_\alpha := \{X\in \mathfrak g_{\mathbb C}: \operatorname{Ad}(t)X=\alpha(t)X\text{ for all }t\in T\} \end{align*} is non-zero. The set of roots is denoted $\Phi\subset X^*(T)$. [/definition] The zero weight space is the complexified Lie algebra of $T$, and the non-zero weight spaces are the root spaces. Because roots are themselves characters, they generate an integral sublattice of the weight lattice. This motivates the root lattice, which records the adjoint group part of the character theory. [definition: Root Lattice] Let $\Phi\subset X^*(T)$ be the root system of $(G,T)$. The root lattice is \begin{align*} Q := \mathbb Z\Phi \subset X^*(T). \end{align*} [/definition] The root lattice lives in $X^*(T)$, but the reflections of the Weyl group require a dual ingredient in $X_*(T)$. Each root gives a rank-one compact subgroup, and the circle in its diagonal torus is the dual object needed to write the reflection formula integrally. [definition: Coroot] For a root $\alpha\in\Phi$, the coroot $\alpha^\vee\in X_*(T)$ is the cocharacter associated to the rank-one subgroup generated by the root spaces $\mathfrak g_\alpha$ and $\mathfrak g_{-\alpha}$, normalized by \begin{align*} \langle \alpha,\alpha^\vee\rangle=2. \end{align*} The set of coroots is denoted $\Phi^\vee\subset X_*(T)$. [/definition] The normalization $\langle \alpha,\alpha^\vee\rangle=2$ is the compact-group version of the familiar coroot $2\alpha/(\alpha,\alpha)$ from abstract root systems. Since the coroots are cocharacters, their integer span is the dual lattice generated by the rank-one subgroups. We need this lattice because weights will be tested against all rank-one subgroup circles at once. [definition: Coroot Lattice] Let $\Phi^\vee\subset X_*(T)$ be the set of coroots. The coroot lattice is \begin{align*} Q^\vee := \mathbb Z\Phi^\vee \subset X_*(T). \end{align*} [/definition] The coroot lattice provides the periods along which weights must have integer values. The pairing with individual coroots is especially important because the rank-one subgroups behave like $SU(2)$ or $SO(3)$, where representation theory is controlled by integral exponents. This motivates the notion of an integral weight. [definition: Integral Weight] Let $\Phi^\vee$ be the coroot system of $(G,T)$. A character $\lambda\in X^*(T)$ is integral with respect to $\Phi^\vee$ if \begin{align*} \langle \lambda,\alpha^\vee\rangle\in\mathbb Z \end{align*} for every $\alpha\in\Phi$. [/definition] For compact groups, every character of $T$ is integral in this sense because it pairs integrally with every cocharacter. In Lie algebra language, this condition detects which linear functionals can occur as weights of finite-dimensional representations. The next theorem proves that no representation can escape this integrality constraint. [quotetheorem:9361] [citeproof:9361] The theorem turns representation theory into lattice arithmetic, but the conclusion depends on using genuine representations of the compact group and genuine coroot circles. If a linear functional is only a Lie-algebra weight and does not exponentiate to a character of the chosen torus, the pairing may satisfy the rank-one integrality conditions while still failing to define a representation of the global group. The standard example is the two-dimensional representation of $\mathfrak{su}(2)_{\mathbb C}\cong\mathfrak{sl}_2(\mathbb C)$: it has highest weight $1$ for the $SU(2)$ torus, but it does not descend to $SO(3)$ because the central element $-I$ acts by $-1$. Thus coroot integrality is necessary for weights of compact-group representations, while descent to a particular quotient imposes the additional condition $\lambda\in X^*(T)$ for that group. The theorem also does not classify the weights that occur in a representation. It gives an obstruction that every occurring weight must pass. In later character formulae, the integers $\langle\lambda,\alpha^\vee\rangle$ measure how far a weight lies from each reflecting hyperplane. The rank-one case is the best guide to how the global form of the group changes the lattice of allowed weights. [example: The SU(2) and SO(3) Lattices] For $SU(2)$, take \begin{align*} T=\{\operatorname{diag}(z,z^{-1}):z\in U(1)\}. \end{align*} For each $k\in\mathbb Z$, define \begin{align*} \lambda_k(\operatorname{diag}(z,z^{-1}))=z^k. \end{align*} If $t_1=\operatorname{diag}(z,z^{-1})$ and $t_2=\operatorname{diag}(w,w^{-1})$, then $t_1t_2=\operatorname{diag}(zw,(zw)^{-1})$, so \begin{align*} \lambda_k(t_1t_2)=(zw)^k=z^kw^k=\lambda_k(t_1)\lambda_k(t_2). \end{align*} Thus $\lambda_k$ is a character, and the standard torus calculation identifies $X^*(T)$ with $\mathbb Z$ by $k\mapsto\lambda_k$. To compute the root, let $E$ be the endomorphism of $\mathbb C^2$ defined by $E(e_1)=0$ and $E(e_2)=e_1$. For $t=\operatorname{diag}(z,z^{-1})$, the inverse satisfies $t^{-1}e_1=z^{-1}e_1$ and $t^{-1}e_2=ze_2$. Hence \begin{align*} (tEt^{-1})(e_1)=tE(z^{-1}e_1)=0. \end{align*} Also, \begin{align*} (tEt^{-1})(e_2)=tE(ze_2)=t(ze_1)=z^2e_1. \end{align*} Since $E(e_1)=0$ and $E(e_2)=e_1$, these two values give \begin{align*} tEt^{-1}=z^2E. \end{align*} Therefore the positive root is $\alpha=\lambda_2$, and the other root is $-\alpha=\lambda_{-2}$. The root lattice is \begin{align*} Q=\mathbb Z\alpha=\mathbb Z\lambda_2\cong 2\mathbb Z\subset X^*(T)\cong\mathbb Z. \end{align*} So for $SU(2)$ the full weight lattice is $\mathbb Z$, while the root lattice is $2\mathbb Z$. Now pass to \begin{align*} SO(3)=SU(2)/\{\pm I\}. \end{align*} A character $\lambda_k$ of $T$ descends to the quotient torus exactly when it is trivial on the kernel $\{\pm I\}\cap T$. Since \begin{align*} -I=\operatorname{diag}(-1,-1)=\operatorname{diag}((-1),(-1)^{-1}), \end{align*} we get \begin{align*} \lambda_k(-I)=(-1)^k. \end{align*} This equals $1$ exactly when $k$ is even. Thus \begin{align*} X^*(T/\{\pm I\})=\{\lambda_k:k\in 2\mathbb Z\}\cong 2\mathbb Z. \end{align*} The root $\lambda_2$ is still present because $\lambda_2(-I)=(-1)^2=1$, so it descends to $SO(3)$. Therefore the $SO(3)$ weight lattice is $2\mathbb Z$, exactly matching the root lattice. [/example] ## Dominant Weights and Fundamental Weights The Weyl group acts on the weight lattice, so the same irreducible representation has many weights related by reflections. The next problem is to choose a unique representative in each Weyl chamber and then identify which representatives classify irreducible representations. [definition: Positive Roots and Simple Roots] Let $\Phi$ be a root system in $X^*(T)\otimes_{\mathbb Z}\mathbb R$. A choice of positive roots is a subset $\Phi^+\subset\Phi$ for which there exists a real linear functional $\eta:X^*(T)\otimes_{\mathbb Z}\mathbb R\to\mathbb R$, non-zero on every root, such that \begin{align*} \Phi^+=\{\alpha\in\Phi:\eta(\alpha)>0\}. \end{align*} The simple roots $\Delta\subset\Phi^+$ are the positive roots that cannot be written as a sum of two positive roots. [/definition] For this choice, exactly one of $\alpha$ and $-\alpha$ lies in $\Phi^+$ for each root $\alpha$, and if $\alpha,\beta\in\Phi^+$ with $\alpha+\beta\in\Phi$, then $\alpha+\beta\in\Phi^+$. Choosing positive roots is equivalent to choosing a Weyl chamber. The simple roots are the inward walls of the chamber, while coroots define the integral inequalities that describe the chamber in lattice terms. This leads to the weights that sit on the chosen non-negative side of every wall. [definition: Dominant Weight] Let $\Delta$ be a set of simple roots. A weight $\lambda\in X^*(T)$ is dominant if \begin{align*} \langle \lambda,\alpha^\vee\rangle\ge 0 \end{align*} for every $\alpha\in\Delta$. [/definition] Dominance gives a partial order on weights and selects the representative expected to label an irreducible representation. To turn that representative into an invariant of a representation, we need the notion of a weight from which all other weights are obtained by subtracting positive roots. That is the highest weight. [definition: Highest Weight] Let $V$ be a finite-dimensional complex representation of $G$. A weight $\lambda$ of $V$ is a highest weight of $V$ if $\lambda+\alpha$ is not a weight of $V$ for every positive root $\alpha\in\Phi^+$. [/definition] For an irreducible representation, this maximality condition has the stronger consequence that every weight lies below the highest weight in the positive-root order. In that case every weight of $V$ has the form \begin{align*} \lambda-\sum_{\alpha\in\Delta} n_\alpha\alpha \end{align*} with $n_\alpha\in\mathbb Z_{\ge 0}$. The highest weight is not merely a label: it determines the irreducible representation. The prior definitions isolate the possible labels, but the key result is that the possible labels are exactly the dominant weights in the character lattice of the chosen global group. This is the central classification theorem for finite-dimensional representations of compact connected Lie groups. [quotetheorem:9730] [proofunderconstruction:9730] This theorem explains why the global lattice matters, and each hypothesis has a role. Irreducibility is needed because a reducible representation can have several incomparable highest weights, one for each irreducible summand, so a single label would not classify it. Compact connectedness is what guarantees complete reducibility, a well-behaved maximal torus, and a finite weight set; without connectedness, extra finite-component data can act on representations without being detected by the root system of the identity component. The choice of positive roots is also part of the labelling convention: changing the chamber replaces the highest weight by a Weyl-conjugate representative, so dominance is meaningful only after that choice. The theorem does not say that every dominant weight for the abstract root system occurs for every global form. For $\mathfrak{su}(2)$, the dominant weights are non-negative integers for the simply connected group $SU(2)$, but only the even ones descend to $SO(3)$. The same root system can therefore have different compact groups sitting over it, and the allowed dominant highest weights are those lying in the character lattice of the chosen group. For calculations, it is useful to express dominant weights in a basis dual to the simple coroots. [definition: Fundamental Weights] Let $\Delta=\{\alpha_1,\dots,\alpha_r\}$ be a basis of simple roots with corresponding coroots $\alpha_1^\vee,\dots,\alpha_r^\vee$. The fundamental weights are the elements $\omega_1,\dots,\omega_r$ of $X^*(T)\otimes_{\mathbb Z}\mathbb R$ defined by \begin{align*} \langle \omega_i,\alpha_j^\vee\rangle=\delta_{ij}. \end{align*} [/definition] When the fundamental weights lie in $X^*(T)$, every dominant integral weight is a non-negative integer combination of them. For non-simply connected quotients, some fundamental weights may not be genuine characters of the maximal torus. Type $A_2$ is the smallest example where two fundamental weights interact in a nonzero way. [example: Fundamental Weights for A2] For type $A_2$, work in the plane \begin{align*} E=\{(x_1,x_2,x_3)\in\mathbb R^3:x_1+x_2+x_3=0\} \end{align*} with roots $e_i-e_j$ and simple roots $\alpha_1=e_1-e_2$, $\alpha_2=e_2-e_3$. With the standard inner product, each root has squared length \begin{align*} (\alpha_1,\alpha_1)=(e_1-e_2,e_1-e_2)=1+1=2. \end{align*} Similarly $(\alpha_2,\alpha_2)=2$, so the coroots are $\alpha_1^\vee=\alpha_1$ and $\alpha_2^\vee=\alpha_2$. We compute the fundamental weights by solving $\langle\omega_i,\alpha_j^\vee\rangle=\delta_{ij}$. Let \begin{align*} \omega_1=\frac{2e_1-e_2-e_3}{3}. \end{align*} Then \begin{align*} \langle\omega_1,\alpha_1^\vee\rangle=\left(\frac{2e_1-e_2-e_3}{3},e_1-e_2\right)=\frac{2+1}{3}=1. \end{align*} Also, \begin{align*} \langle\omega_1,\alpha_2^\vee\rangle=\left(\frac{2e_1-e_2-e_3}{3},e_2-e_3\right)=\frac{-1+1}{3}=0. \end{align*} Now let \begin{align*} \omega_2=\frac{e_1+e_2-2e_3}{3}. \end{align*} Then \begin{align*} \langle\omega_2,\alpha_1^\vee\rangle=\left(\frac{e_1+e_2-2e_3}{3},e_1-e_2\right)=\frac{1-1}{3}=0. \end{align*} And \begin{align*} \langle\omega_2,\alpha_2^\vee\rangle=\left(\frac{e_1+e_2-2e_3}{3},e_2-e_3\right)=\frac{1+2}{3}=1. \end{align*} Thus these two vectors satisfy the defining equations for the fundamental weights. The root lattice is generated by \begin{align*} \alpha_1=e_1-e_2 \end{align*} and \begin{align*} \alpha_2=e_2-e_3. \end{align*} For $SU(3)$, the simply connected weight lattice contains $\omega_1$ and $\omega_2$, so they are genuine weights. For the adjoint form $PSU(3)$, only characters in the root lattice descend; $\omega_1$ and $\omega_2$ pair integrally with coroots but are not themselves integer combinations of $\alpha_1$ and $\alpha_2$. Thus $SU(3)$ sees the full weight lattice, while $PSU(3)$ keeps only the root lattice characters. [/example] ## Center and Finite Quotients from Lattice Inclusions The local Lie algebra determines the root system, but compact connected groups with the same Lie algebra can differ by finite central quotients. The lattice question is therefore: which intermediate character lattices are allowed between the root lattice and the full simply connected weight lattice? [definition: Simply Connected Weight Lattice] Let $\Phi$ be a root system with coroots $\Phi^\vee$. The simply connected weight lattice is \begin{align*} P := \{\lambda\in X^*(T)\otimes_{\mathbb Z}\mathbb R : \langle\lambda,\alpha^\vee\rangle\in\mathbb Z\text{ for all }\alpha\in\Phi\}. \end{align*} [/definition] This lattice is the character lattice for a maximal torus in the simply connected compact group with the given semisimple Lie algebra. At the other extreme, the adjoint group has character lattice equal to the root lattice. We now need the main structural theorem saying that every compact semisimple global form sits between these two extremes. [quotetheorem:9731] [citeproof:9731] The quotient $P/Q$ is therefore the character group of the center of the simply connected group. The semisimplicity hypothesis is essential: it makes the center finite and forces the root lattice to span the relevant real vector space. If a central torus factor is present, roots do not see it. For instance, $U(n)$ has the same semisimple root system as $SU(n)$ in the traceless directions, but the determinant character supplies an additional infinite cyclic direction in $X^*(T)$ that is not squeezed between the type $A_{n-1}$ root lattice and its simply connected weight lattice. Thus the theorem classifies only the semisimple global forms for a fixed root system. It does not classify compact connected groups with torus factors, and it does not by itself identify the multiplication law between a central torus and the semisimple part. Those data are packaged later by the full root datum. The classical unitary groups show how the semisimple and central torus parts appear in practice. [example: SU(n), PSU(n), and U(n)] For $SU(n)$, take the maximal torus \begin{align*} T_{SU}=\{\operatorname{diag}(z_1,\dots,z_n):z_i\in U(1),\ z_1\cdots z_n=1\}. \end{align*} If $a=(a_1,\dots,a_n)\in\mathbb Z^n$, the formula \begin{align*} \lambda_a(\operatorname{diag}(z_1,\dots,z_n))=z_1^{a_1}\cdots z_n^{a_n} \end{align*} defines a character of $T_{SU}$. The exponent vectors $a$ and $a+c(1,\dots,1)$ define the same character, because on $T_{SU}$ we have \begin{align*} z_1^{a_1+c}\cdots z_n^{a_n+c}=z_1^{a_1}\cdots z_n^{a_n}(z_1\cdots z_n)^c=z_1^{a_1}\cdots z_n^{a_n}. \end{align*} Thus \begin{align*} X^*(T_{SU})\cong \mathbb Z^n/\mathbb Z(1,\dots,1), \end{align*} which is the simply connected weight lattice of type $A_{n-1}$. The roots are obtained from the adjoint action on matrix units. Let $E_{ij}$ be the matrix with $1$ in the $(i,j)$ entry and $0$ elsewhere. For $t=\operatorname{diag}(z_1,\dots,z_n)$ and the standard basis vector $e_k$, one has $t^{-1}e_j=z_j^{-1}e_j$, so \begin{align*} (tE_{ij}t^{-1})(e_j)=tE_{ij}(z_j^{-1}e_j)=t(z_j^{-1}e_i)=z_i z_j^{-1}e_i. \end{align*} If $k\ne j$, then $E_{ij}(t^{-1}e_k)=0$, so $tE_{ij}t^{-1}$ agrees with $(z_i z_j^{-1})E_{ij}$ on every basis vector. Hence the root acting on $E_{ij}$ is \begin{align*} \alpha_{ij}=e_i-e_j. \end{align*} The root lattice is therefore \begin{align*} Q=\mathbb Z\{e_i-e_j: i\ne j\}\subset \mathbb Z^n/\mathbb Z(1,\dots,1). \end{align*} Now pass to $PSU(n)=SU(n)/\mu_n$, where \begin{align*} \mu_n=\{\zeta I:\zeta^n=1\}. \end{align*} A character $\lambda_a$ descends to $PSU(n)$ exactly when it is trivial on $\mu_n$. For $\zeta^n=1$, \begin{align*} \lambda_a(\zeta I)=\zeta^{a_1}\cdots \zeta^{a_n}=\zeta^{a_1+\cdots+a_n}. \end{align*} This equals $1$ for every $n$th root of unity $\zeta$ exactly when \begin{align*} a_1+\cdots+a_n\equiv 0 \pmod n. \end{align*} If this congruence holds, write $a_1+\cdots+a_n=nm$ and replace $a$ by \begin{align*} b=a-m(1,\dots,1). \end{align*} This replacement does not change the character on $T_{SU}$, and \begin{align*} b_1+\cdots+b_n=(a_1+\cdots+a_n)-nm=0. \end{align*} For any such $b$, set $c_k=b_1+\cdots+b_k$ for $1\le k\le n-1$. Then \begin{align*} \sum_{k=1}^{n-1}c_k(e_k-e_{k+1})=b, \end{align*} because the first coordinate is $c_1=b_1$, the $i$th coordinate for $2\le i\le n-1$ is $-c_{i-1}+c_i=b_i$, and the last coordinate is $-c_{n-1}=b_n$. Hence the descended character lattice is exactly the root lattice $Q$. For $U(n)$, the diagonal torus is \begin{align*} T_U=(U(1))^n, \end{align*} so \begin{align*} X^*(T_U)\cong \mathbb Z^n \end{align*} with no relation $z_1\cdots z_n=1$. The same computation with $E_{ij}$ gives roots $e_i-e_j$, while the determinant character is \begin{align*} \det(\operatorname{diag}(z_1,\dots,z_n))=z_1\cdots z_n, \end{align*} corresponding to the exponent vector $(1,\dots,1)$. This vector is central and is not generated by the differences $e_i-e_j$, whose coordinate sums are all $0$. Thus $U(n)$ has the same type $A_{n-1}$ semisimple roots as $SU(n)$, together with one additional infinite cyclic central weight direction coming from the determinant. [/example] The examples show that the root system alone cannot distinguish simply connected and adjoint forms, and it also omits central torus directions. The final invariant needs to package the character lattice, cocharacter lattice, roots, and coroots together. This motivates root data, which remember both the semisimple Lie algebra and the finite global information. [definition: Root Datum] A root datum is a quadruple \begin{align*} (X^*,\Phi,X_*,\Phi^\vee) \end{align*} where $X^*$ and $X_*$ are finite-rank free abelian groups in perfect duality, $\Phi\subset X^*$, $\Phi^\vee\subset X_*$, and there is a bijection $\alpha\mapsto\alpha^\vee$ from $\Phi$ to $\Phi^\vee$ satisfying the usual root reflection axioms \begin{align*} s_\alpha:X^* \to X^*, \qquad s_\alpha(\lambda)=\lambda-\langle\lambda,\alpha^\vee\rangle\alpha. \end{align*} The dual reflection is \begin{align*} s_{\alpha^\vee}:X_* \to X_*, \qquad s_{\alpha^\vee}(\mu)=\mu-\langle\alpha,\mu\rangle\alpha^\vee. \end{align*} [/definition] The root datum retains the roots and coroots, but also keeps the ambient lattices. This is what distinguishes $SU(2)$ from $SO(3)$ even though their complexified Lie algebras are both $\mathfrak{sl}_2(\mathbb C)$. The following theorem states that, for compact connected groups, this is exactly the right amount of information. [quotetheorem:9733] This classification is quoted here as the structural endpoint of the chapter. Its construction combines the classification of compact semisimple Lie algebras by root systems with the lattice description of finite central quotients developed above, and then incorporates the central torus through the ambient character and cocharacter lattices. The compact connected hypothesis is essential: a disconnected compact group such as $O(2)$ has identity component $SO(2)$ with the same torus lattice as a circle, but the extra component acts by inversion and is not recorded by the connected root datum. Outside compactness, the additive Lie group $\mathbb R$ has the same Lie algebra as the circle locally, but its continuous characters are parametrized by all real frequencies rather than by an integral lattice. [remark: What the Root Datum Remembers] The root system alone remembers the Weyl reflections and the semisimple Lie algebra. The root datum remembers which weights are honest characters of the maximal torus, which cocharacters are honest circles in the torus, and how roots pair with coroots. Thus it records both the infinitesimal structure and the finite central information needed to reconstruct the compact connected group. [/remark] Root data converts the geometry of maximal tori into a combinatorial invariant that can distinguish compact connected Lie groups. With that classification data in hand, the course turns to highest weight theory, where irreducible representations are labeled and recovered from finite sets of weights. # 6. Highest Weight Theory for Compact Groups Highest weight theory turns the structure of a compact connected Lie group into a problem about finite sets of weights. In the preceding chapters, maximal tori, Weyl groups, and root systems explained how conjugacy and the adjoint action are controlled by a torus. The next question is how an arbitrary finite-dimensional complex representation is controlled by the same torus, and which torus weights can occur at the top of an irreducible representation. The central result is the highest weight theorem: irreducible finite-dimensional complex representations of a compact connected Lie group are classified by dominant integral weights. The proof combines complete reducibility, the root-space decomposition of the complexified Lie algebra, and the existence of vectors killed by all positive root operators. This chapter develops the language needed to state the theorem and then records its main examples for $SU(2)$, $SU(3)$, and $SU(n)$. ## Weight Decompositions of Representations A representation of a compact connected group $G$ may be complicated on $G$ itself, but its restriction to a maximal torus $T$ is simultaneously diagonalizable. The first problem is therefore to describe a representation by the characters through which $T$ acts, and then to understand how the rest of $G$ moves between the corresponding eigenspaces. [definition: Weight Of A Torus Representation] Let $T$ be a compact torus, let $V$ be a finite-dimensional complex representation of $T$, and let $\nu:T\to U(1)$ be a continuous character. The $\nu$-weight space of $V$ is \begin{align*} V_\nu:=\{v\in V:t\cdot v=\nu(t)v\text{ for all }t\in T\}. \end{align*} A character $\nu$ is a weight of $V$ if $V_\nu\ne 0$. [/definition] The weight space is the common eigenspace for the commuting family of operators $\rho(t)$, where $t\in T$. This definition gives names to the possible torus eigenspaces, but it does not yet say that they exhaust the representation. The next theorem is needed because highest weight theory only works once every vector can be decomposed into weight components. [quotetheorem:9726] [citeproof:9726] This theorem says that the torus part of a representation is completely discrete: it is a finite multiset of characters, counted with multiplicity. Compactness is essential here, since representations of non-compact abelian groups need not be unitary after averaging and may contain Jordan blocks rather than eigenspace decompositions. [Finite dimensionality](/theorems/1534) is also part of the conclusion: infinite-dimensional torus representations can involve infinitely many weights or continuous spectral phenomena. To compare these characters with roots, we need to replace characters of $T$ by differentials on $\mathfrak t$, so that weights and roots live in the same vector space. [definition: Integral Weight Lattice] Let $G$ be a compact connected Lie group, let $T\subset G$ be a maximal torus, and let $\mathfrak t=\operatorname{Lie}(T)$. For a character $\nu\in X^*(T)$, its differential at the identity is a real-linear map \begin{align*} d\nu_e:\mathfrak t\to \operatorname{Lie}(U(1))=i\mathbb R. \end{align*} The integral weight lattice is \begin{align*} \Lambda:=\{\lambda\in i\mathfrak t^*: \lambda=d\nu_e\text{ for some }\nu\in X^*(T)\}. \end{align*} [/definition] After identifying a character with its differential, the weights of any finite-dimensional representation of $G$ lie in $\Lambda$. The roots occur as torus weights of the adjoint representation, so for the adjoint group action they are genuine characters of $T$ and hence give elements of $X^*(T)$ in this sense. The global-topology issue is different: the full lattice of possible highest weights for arbitrary representations depends on $G$ itself, and may be smaller than the weight lattice of the simply connected group with the same Lie algebra. [example: Weights Of The Standard Representation Of SU(n)] Let $G=SU(n)$ and let \begin{align*} T=\{\operatorname{diag}(t_1,\dots,t_n): t_j\in U(1)\text{ and }t_1\cdots t_n=1\}. \end{align*} On the standard representation $V=\mathbb C^n$, write $e_1,\dots,e_n$ for the standard basis. For $t=\operatorname{diag}(t_1,\dots,t_n)\in T$, matrix multiplication gives \begin{align*} t e_j=\operatorname{diag}(t_1,\dots,t_n)(0,\dots,1,\dots,0)^\top=t_j e_j. \end{align*} Thus the coordinate line $\mathbb C e_j$ is a weight space for the character $\varepsilon_j:T\to U(1)$ defined by \begin{align*} \varepsilon_j(\operatorname{diag}(t_1,\dots,t_n))=t_j. \end{align*} Every vector $v\in\mathbb C^n$ has a unique expansion \begin{align*} v=c_1e_1+\cdots+c_ne_n. \end{align*} Applying $t$ gives \begin{align*} t v=c_1t_1e_1+\cdots+c_nt_ne_n. \end{align*} So the only possible torus eigencharacters appearing in the standard basis decomposition are $\varepsilon_1,\dots,\varepsilon_n$, and each line $\mathbb C e_j$ is one-dimensional. Hence the weights of the standard representation are $\varepsilon_1,\dots,\varepsilon_n$, each with multiplicity one. On the Lie algebra \begin{align*} \mathfrak t=\{\operatorname{diag}(i\theta_1,\dots,i\theta_n):\theta_1+\cdots+\theta_n=0\}, \end{align*} the differentials satisfy \begin{align*} (d\varepsilon_1+\cdots+d\varepsilon_n)(\operatorname{diag}(i\theta_1,\dots,i\theta_n))=i(\theta_1+\cdots+\theta_n)=0. \end{align*} This is the relation usually written $\varepsilon_1+\cdots+\varepsilon_n=0$ on $\mathfrak t$. The determinant-one condition is exactly why the coordinate weights are not independent; nevertheless, for a diagonal torus action the weight on each coordinate line is read off from the scalar multiplying that coordinate. [/example] The example shows why weights are best treated as elements of a lattice modulo the determinant-one relation. Listing the weights still leaves an important problem unresolved: the non-torus part of $G$ must act somewhere, and the root decomposition predicts exactly which weight spaces it can connect. The next theorem supplies this bridge between roots and representation weights. [quotetheorem:9362] [citeproof:9362] Thus the roots act as moves between weight spaces. The connectedness of $T$ matters in passing from infinitesimal weights back to torus characters; without it, differential data can miss finite component characters. The finite-dimensional hypothesis is what turns this shifting rule into a highest-weight argument, because only finitely many weights can be reached before the raising process stops. After choosing a direction in which some roots count as positive, there must therefore be weights which cannot be raised further. ## Positive Roots and Highest Weight Vectors The next problem is to choose an ordering of the weight lattice that is compatible with the root system. Once positive roots are chosen, a highest weight vector is a vector whose weight cannot be increased by any positive root operator. This reduces irreducible representations to cyclic modules generated from a single top vector. [definition: Positive Root System] Let $R\subset i\mathfrak t^*$ be the root system of $(G,T)$. A positive root system is a subset $R^+\subset R$ such that $R=R^+\sqcup(-R^+)$ and whenever $\alpha,\beta\in R^+$ with $\alpha+\beta\in R$, then $\alpha+\beta\in R^+$. [/definition] The choice of $R^+$ is the representation-theoretic analogue of choosing upper triangular matrices. It tells us which root operators raise weights, but to compare arbitrary weights we need a precise order generated by the simple positive roots. [definition: Dominance Order] Let $R^+$ be a positive root system with simple roots $\Delta=\{\alpha_1,\dots,\alpha_r\}$. For weights $\mu,\lambda\in i\mathfrak t^*$, write $\mu\le\lambda$ if \begin{align*} \lambda-\mu=\sum_{j=1}^r n_j\alpha_j \end{align*} for some integers $n_j\ge 0$. [/definition] Because every finite-dimensional representation has finitely many weights, this order always has maximal elements among the weights that occur. A maximal weight is expected to be at the top of the representation, but the order alone does not record how positive root spaces act on actual vectors. The next definition is needed to isolate vectors killed by all raising operators, which is the condition that lets the entire representation be recovered by lowering. [definition: Highest Weight Vector] Let $\rho:G\to GL(V)$ be a finite-dimensional complex representation of $G$, and let $d\rho:\mathfrak g_\mathbb C\to \mathfrak{gl}(V)$ be its derived complex-linear representation. Choose $R^+$. A non-zero vector $v\in V_\lambda$ is a highest weight vector of weight $\lambda$ if \begin{align*} d\rho(X)v=0\quad\text{for all }X\in\mathfrak g_\alpha\text{ and all }\alpha\in R^+. \end{align*} [/definition] The annihilation condition is stronger in form than saying that $\lambda$ is maximal among the weights, but the root-shifting theorem shows that a maximal weight vector satisfies it. The next result is the first structural payoff: every non-zero representation contains at least one vector from which a highest-weight analysis can begin. [quotetheorem:9734] [citeproof:9734] This proves existence without using irreducibility. The result would fail as stated without finite dimensionality, since an infinite set of weights may have no maximal element in the dominance order. It also explains why a choice of positive roots is not cosmetic: changing the chamber changes which extremal weight is selected. For classification, existence is not enough: if an irreducible representation could have several unrelated top weights or a multi-dimensional top space, a highest weight would not be a label. The next theorem rules out both possibilities. [quotetheorem:9736] [citeproof:9736] The one-dimensionality of the highest weight space is what makes classification possible. Irreducibility is essential: a direct sum of two copies of the same irreducible has a two-dimensional highest weight space, and a direct sum of different irreducibles can have several incomparable maximal weights. The PBW argument also has a precise scope. It orders weights only by subtracting roots from the semisimple part of $\mathfrak g_\mathbb C$; central torus directions act by scalars and do not create raising or lowering operators. The theorem therefore identifies the top label and the fact that all other weights lie below it in root directions, but it does not compute the multiplicities of lower weights. To state which top labels can occur, we need the integrality and positivity conditions measured against coroots. [definition: Dominant Integral Weight] Let $R^+$ be a positive root system with simple roots $\Delta$. For each root $\alpha$, let $\alpha^\vee$ denote the associated coroot, viewed as an element of $(i\mathfrak t^*)^*$ so that the pairing $\langle\lambda,\alpha^\vee\rangle$ is defined for $\lambda\in i\mathfrak t^*$. A weight $\lambda\in\Lambda$ is dominant if \begin{align*} \langle \lambda,\alpha^\vee\rangle\ge 0 \end{align*} for every simple root $\alpha\in\Delta$. A dominant integral weight means a dominant element of the integral weight lattice $\Lambda$. [/definition] Dominance is exactly the condition imposed by the embedded $SU(2)$-subgroups attached to simple roots. Restricting a representation to such a subgroup gives the familiar fact that the highest weight for $SU(2)$ must be a non-negative integer. [example: Highest Weights For SU(2)] For $G=SU(2)$, take \begin{align*} T=\{\operatorname{diag}(z,z^{-1}):z\in U(1)\}. \end{align*} The character lattice is indexed by integers: the integer $k$ denotes the character $\chi_k(\operatorname{diag}(z,z^{-1}))=z^k$. On the standard basis of $\mathbb C^2$, \begin{align*} \operatorname{diag}(z,z^{-1})e_1=ze_1 \end{align*} and \begin{align*} \operatorname{diag}(z,z^{-1})e_2=z^{-1}e_2. \end{align*} Thus $e_1$ has weight $1$ and $e_2$ has weight $-1$. A basis of $\operatorname{Sym}^m(\mathbb C^2)$ is given by the monomials \begin{align*} e_1^{m-r}e_2^r,\qquad 0\le r\le m. \end{align*} For $t=\operatorname{diag}(z,z^{-1})$, the induced action on symmetric tensors gives \begin{align*} t\cdot e_1^{m-r}e_2^r=(ze_1)^{m-r}(z^{-1}e_2)^r=z^{m-r}z^{-r}e_1^{m-r}e_2^r=z^{m-2r}e_1^{m-r}e_2^r. \end{align*} So the weights are \begin{align*} m,m-2,m-4,\dots,-m, \end{align*} and each occurs with multiplicity one because each value $m-2r$ comes from exactly one basis vector $e_1^{m-r}e_2^r$. With the usual positive root chosen to have weight $2$, the largest weight in this list is $m$. It is represented by $e_1^m$, since \begin{align*} t\cdot e_1^m=(ze_1)^m=z^m e_1^m. \end{align*} The raising operator sends $e_1$ to $0$ and $e_2$ to a scalar multiple of $e_1$, so it kills $e_1^m$ and raises every other monomial by replacing one factor of $e_2$ with $e_1$. Hence $e_1^m$ is the highest weight vector. The highest weight classification for $SU(2)$ says that every irreducible finite-dimensional complex representation is obtained in this way, so the irreducibles are exactly $\operatorname{Sym}^m(\mathbb C^2)$ for $m\ge0$. [/example] The $SU(2)$ example is the local model for all simple roots. If a proposed highest weight has negative pairing with a simple coroot, then the corresponding rank-one $SU(2)$ string would have negative length, which is impossible in a finite-dimensional representation. Each simple root direction therefore cuts out a concrete obstruction, not merely a formal inequality, and the non-negativity of all these string lengths is precisely the dominance condition. ## The Highest Weight Theorem We now ask the classification question: if $\lambda$ is a dominant integral weight, does there exist an irreducible representation with highest weight $\lambda$, and is it unique? The answer is affirmative for compact connected groups once the weight lattice is the actual character lattice of the chosen group, not merely the lattice of the simply connected cover. [quotetheorem:9737] [proofunderconstruction:9737] The theorem packages the classification into lattice combinatorics, but three conditions are being combined. Dominance is the chamber condition $\langle\lambda,\alpha^\vee\rangle\ge0$, which comes from the simple-root $SU(2)$ strings. Integrality against coroots is the Lie-algebra condition that those strings have integer lengths. Membership in the actual character lattice $X^*(T)$ is the global group condition; it determines whether the representation descends from the simply connected cover to $G$. For example, $SU(2)$ and $SO(3)$ have the same Lie algebra and the same root system, but only the even highest weights of $SU(2)$ descend to representations of $SO(3)$. This is the main limitation of a purely Lie-algebraic statement and the reason the character lattice appears explicitly. Once $R^+$ is fixed, irreducibles are labelled by the lattice points of $X^*(T)$ lying in the closed Weyl chamber cut out by the dominance inequalities. [remark: Dependence On The Choice Of Positive Roots] Changing $R^+$ changes which weight is called highest, but not the representation. The Weyl group acts simply transitively on the Weyl chambers, so the highest weight with respect to a new chamber is the corresponding Weyl group translate of the old extremal weight. [/remark] The classification theorem is useful because standard operations on representations have transparent effects on weights. We next record the three operations used throughout character theory: direct sums and tensor products, duals, and exterior powers. [quotetheorem:9738] [citeproof:9738] [Tensor product](/page/Tensor%20Product) weights give candidate highest weights for irreducible summands, but the tensor product need not be irreducible. The theorem is only a statement about the restricted torus action: it does not determine which irreducible summands occur or their multiplicities as $G$-representations. For instance, for $SU(2)$ the tensor product of two irreducibles has weights obtained by addition, but these weights must still be reorganised into the Clebsch-Gordan summands. The largest possible highest weight is the sum of the two highest weights, and lower summands are constrained by the same dominance order. The hypotheses are doing real work. Finite dimensionality ensures that $V$ and $W$ split as finite direct sums of torus weight spaces, so the tensor product is the algebraic direct sum of the spaces $V_\lambda\otimes W_\mu$. In an infinite-dimensional representation of $S^1$ on $L^2(S^1)$ by rotation, the Fourier characters give infinitely many weight spaces; after tensoring two such representations, multiplicities at a fixed weight can be infinite because there are infinitely many pairs of integers with a given sum. The representation hypothesis is also necessary: for two vector spaces with arbitrary linear $T$-actions that do not respect the group law, there is no character multiplication rule $\lambda(t)\mu(t)$. Thus the formula belongs to finite-dimensional torus-diagonalizable representations, not to arbitrary linear operators on vector spaces. The next operation is duality, where the torus action is inverted rather than multiplied, so every weight should be reflected through the origin. [quotetheorem:9739] [citeproof:9739] For an irreducible representation of highest weight $\lambda$, the highest weight of the dual is not usually $-\lambda$, because $-\lambda$ is typically antidominant. This is a useful warning that the dual-weight theorem records the whole torus weight multiset, not the highest-weight label directly. The Weyl chamber must be applied after dualising: the dominant highest weight of $V^*$ is $-w_0\lambda$, where $w_0$ is the longest element of the Weyl group. Here again the finite-dimensional representation hypothesis is essential. The proof identifies $V^*$ with the direct sum of dual weight spaces, which is valid for finite direct sums. If $V=\bigoplus_{n\in\mathbb Z}\mathbb C e_n$ is an infinite-dimensional algebraic representation of $S^1$ with weights $n$, then the full algebraic dual $V^*$ is a product of the dual lines, not a direct sum; it contains functionals with infinitely many non-zero components, so it is not exhausted by weight vectors of weights $-n$. If the original action is not a genuine group representation, the contragredient formula $(t\cdot f)(v)=f(t^{-1}\cdot v)$ need not define a group action. The theorem therefore describes duals inside the finite-dimensional representation category. [example: Standard Dual And Adjoint Representations Of SU(3)] For $SU(3)$, take \begin{align*} T=\{\operatorname{diag}(t_1,t_2,t_3):t_j\in U(1)\text{ and }t_1t_2t_3=1\}, \end{align*} and write $\varepsilon_j(\operatorname{diag}(t_1,t_2,t_3))=t_j$. Since $t_1t_2t_3=1$, the corresponding differentials satisfy $\varepsilon_1+\varepsilon_2+\varepsilon_3=0$ on $\mathfrak t$. With positive roots chosen as $\varepsilon_i-\varepsilon_j$ for $i<j$, the standard representation has weights $\varepsilon_1,\varepsilon_2,\varepsilon_3$, and $\varepsilon_1$ is highest because \begin{align*} \varepsilon_1-\varepsilon_2=\alpha_1 \end{align*} and \begin{align*} \varepsilon_1-\varepsilon_3=(\varepsilon_1-\varepsilon_2)+(\varepsilon_2-\varepsilon_3)=\alpha_1+\alpha_2. \end{align*} For the dual representation, the action is $(t\cdot f)(v)=f(t^{-1}v)$. If $e_j^*$ is dual to $e_j$, then \begin{align*} (t\cdot e_j^*)(e_j)=e_j^*(t^{-1}e_j)=e_j^*(t_j^{-1}e_j)=t_j^{-1}. \end{align*} Thus $e_j^*$ has weight $-\varepsilon_j$, so the dual weights are $-\varepsilon_1,-\varepsilon_2,-\varepsilon_3$. Among these, $-\varepsilon_3$ is highest because \begin{align*} (-\varepsilon_3)-(-\varepsilon_2)=\varepsilon_2-\varepsilon_3=\alpha_2 \end{align*} and \begin{align*} (-\varepsilon_3)-(-\varepsilon_1)=\varepsilon_1-\varepsilon_3=\alpha_1+\alpha_2. \end{align*} For the adjoint representation, complexify to $\mathfrak{sl}_3(\mathbb C)$. Let $E_{ij}$ be the matrix with $1$ in the $(i,j)$ entry and $0$ elsewhere. For $i\ne j$, \begin{align*} \operatorname{Ad}_t(E_{ij})=tE_{ij}t^{-1}=t_i t_j^{-1}E_{ij}. \end{align*} Therefore $E_{ij}$ has weight $\varepsilon_i-\varepsilon_j$. The diagonal trace-zero subspace $\mathfrak t_\mathbb C$ is fixed by conjugation, since $tHt^{-1}=H$ for every diagonal $H$, so it contributes the zero weight with multiplicity $\dim\mathfrak t_\mathbb C=2$. Hence the adjoint weights are the six roots $\varepsilon_i-\varepsilon_j$ for $i\ne j$, together with zero of multiplicity $2$. The highest adjoint weight is $\varepsilon_1-\varepsilon_3$. Indeed, \begin{align*} (\varepsilon_1-\varepsilon_3)-(\varepsilon_1-\varepsilon_2)=\varepsilon_2-\varepsilon_3=\alpha_2 \end{align*} and \begin{align*} (\varepsilon_1-\varepsilon_3)-(\varepsilon_2-\varepsilon_3)=\varepsilon_1-\varepsilon_2=\alpha_1. \end{align*} The remaining non-zero roots are lower still because subtracting a negative root adds a positive root combination. Thus the standard, dual, and adjoint representations have highest weights $\varepsilon_1$, $-\varepsilon_3$, and $\varepsilon_1-\varepsilon_3$, respectively. [/example] This example already shows the pattern for $SU(n)$: standard representations generate many fundamental examples, and exterior powers produce the fundamental highest weights. The boundary to keep in mind is that the standard representation alone only sees the first fundamental weight; to reach the other walls of the dominant chamber one must combine coordinate weights in alternating products. Exterior powers are the cleanest way to do this because their weight vectors are indexed by subsets of coordinate lines. [example: Exterior Powers Of The Standard Representation Of SU(n)] Let $V=\mathbb C^n$ be the standard representation of $SU(n)$, with standard basis $e_1,\dots,e_n$ and torus weights $\varepsilon_1,\dots,\varepsilon_n$. Fix $1\le k\le n$. The exterior power $\Lambda^k V$ has basis vectors \begin{align*} e_{i_1}\wedge\cdots\wedge e_{i_k}\qquad 1\le i_1<\cdots<i_k\le n. \end{align*} For $t=\operatorname{diag}(t_1,\dots,t_n)\in T$, the induced action on exterior powers is obtained by applying $t$ to each factor: \begin{align*} t\cdot(e_{i_1}\wedge\cdots\wedge e_{i_k})=(t e_{i_1})\wedge\cdots\wedge(t e_{i_k}). \end{align*} Since $t e_{i_j}=t_{i_j}e_{i_j}$ for each $j$, multilinearity of the wedge product gives \begin{align*} (t e_{i_1})\wedge\cdots\wedge(t e_{i_k})=(t_{i_1}e_{i_1})\wedge\cdots\wedge(t_{i_k}e_{i_k})=(t_{i_1}\cdots t_{i_k})(e_{i_1}\wedge\cdots\wedge e_{i_k}). \end{align*} Thus $e_{i_1}\wedge\cdots\wedge e_{i_k}$ has weight $\varepsilon_{i_1}+\cdots+\varepsilon_{i_k}$, because \begin{align*} (\varepsilon_{i_1}+\cdots+\varepsilon_{i_k})(t)=\varepsilon_{i_1}(t)\cdots\varepsilon_{i_k}(t)=t_{i_1}\cdots t_{i_k}. \end{align*} Take the standard positive roots $\varepsilon_a-\varepsilon_b$ for $a<b$. We show that the largest weight among the weights of $\Lambda^k V$ is \begin{align*} \omega_k=\varepsilon_1+\cdots+\varepsilon_k. \end{align*} Let $I=\{i_1<\cdots<i_k\}$ be any $k$-element subset of $\{1,\dots,n\}$, and write \begin{align*} \lambda_I=\varepsilon_{i_1}+\cdots+\varepsilon_{i_k}. \end{align*} If $I=\{1,\dots,k\}$, then $\lambda_I=\omega_k$. Otherwise choose the smallest $r$ such that $r\notin I$. Since $I$ has $k$ elements and is not $\{1,\dots,k\}$, there is some $s\in I$ with $s>r$. Replacing $s$ by $r$ changes the weight by \begin{align*} (\lambda_I-\varepsilon_s+\varepsilon_r)-\lambda_I=\varepsilon_r-\varepsilon_s. \end{align*} Because $r<s$, this difference is a positive root. Repeating this replacement finitely many times turns $I$ into $\{1,\dots,k\}$, so $\omega_k-\lambda_I$ is a sum of positive roots. Hence every weight of $\Lambda^kV$ lies below $\omega_k$ in the dominance order, and $\omega_k$ is the highest weight. For $1\le k\le n-1$, these highest weights $\omega_k=\varepsilon_1+\cdots+\varepsilon_k$ are the fundamental highest weights of $SU(n)$. Thus the exterior powers $\Lambda^k\mathbb C^n$ realize the fundamental representations in the standard $SU(n)$ family. [/example] The exterior power examples provide the building blocks for many irreducibles, though tensor products of fundamental representations must still be decomposed. The highest weight theorem says that every irreducible appears with a unique dominant highest weight, while later character formulae explain the multiplicities and dimensions. ## Complete Reducibility And Consequences The classification by highest weights assumes that finite-dimensional representations can be split into irreducibles. For compact groups, this is not a separate hypothesis but a consequence of averaging inner products over Haar measure. [quotetheorem:8826] [citeproof:8826] Complete reducibility lets us read an arbitrary representation as a finite multiset of dominant highest weights. Compactness is indispensable in this proof because Haar averaging creates the invariant Hermitian inner product; for non-compact groups, finite-dimensional representations need not split into irreducibles. Finite dimensionality is also used in the induction, while infinite-dimensional unitary representation theory requires additional analytic hypotheses. This is the representation-theoretic counterpart of diagonalising a normal operator: irreducibles replace eigenspaces, and highest weights replace eigenvalues. [explanation: How To Use The Classification] To analyse a concrete representation, first restrict it to the maximal torus and list its weights with multiplicities. Next choose positive roots and identify the maximal weights that are dominant. These maximal weights are candidates for highest weights of irreducible summands. Subtract the full weight multiset of the corresponding irreducible, then repeat until all weights have been accounted for. For low-rank groups this procedure is often enough by itself. For higher-rank groups, the Weyl character formula and Freudenthal [multiplicity formula](/theorems/2420) give systematic ways to determine the full weight multiset of an irreducible representation from its highest weight. [/explanation] The next chapter turns this classification into character theory. Characters remember weight multiplicities on the maximal torus, and Weyl group symmetry then controls class functions on the whole group. Highest weight theory completes the classification of irreducible representations, but it also sets up the passage to harmonic analysis on the group. The next chapter translates highest weights into characters and matrix coefficients, and then uses Peter-Weyl theory to describe how those representations fill out `L^2(G)`. # 7. Peter-Weyl Theorem and Character Orthogonality This chapter turns the finite-dimensional representation theory developed earlier into harmonic analysis on a compact Lie group $G$. The central question is how much of $L^2(G)$ is generated by the finite-dimensional representations of $G$, viewed through their matrix entries. The answer is the Peter-Weyl theorem: matrix coefficients form a dense orthogonal algebra of functions, and the characters of irreducible representations form the Fourier basis for class functions. Chapters 2 through 6 used maximal tori, Weyl groups, roots, and weights to organize irreducible representations. Here those representations are assembled back into functions on the group. This passage is the bridge from structure theory to the Weyl integration and character formulae: before characters can be computed, we need to know that characters are the right coordinates on conjugation-invariant functions. ## Matrix Coefficients as Functions on a Compact Group How does a representation of $G$ produce a function on $G$? If $\rho: G \to GL(V)$ is finite-dimensional, then each entry of the matrix of $\rho(g)$ is a complex-valued function of $g$. The coordinate-free form uses a vector in $V$ and a linear functional on $V$. [definition: Matrix Coefficient] Let $G$ be a compact Lie group and let $(\rho,V)$ be a finite-dimensional complex representation of $G$. For $v \in V$ and $\lambda \in V^*$, the associated matrix coefficient is the function $c_{\lambda,v}:G\to \mathbb C$ defined by \begin{align*} c_{\lambda,v}(g)=\lambda(\rho(g)v). \end{align*} [/definition] Matrix coefficients are the basic waves on $G$. When $G$ is abelian they reduce to characters of one-dimensional representations, but for non-abelian $G$ the individual entries of higher-dimensional representations are needed before taking traces. [example: Matrix Coefficients on the Circle] Let $G=\mathbb T=\{z\in \mathbb C: |z|=1\}$. For each $n\in\mathbb Z$, the one-dimensional representation $\rho_n:\mathbb T\to GL(\mathbb C)$ is given by \begin{align*} \rho_n(z)w=z^n w. \end{align*} Taking $v=1\in\mathbb C$ and $\lambda\in(\mathbb C)^*$ defined by $\lambda(w)=w$, the associated matrix coefficient is \begin{align*} c_{\lambda,1}(z)=\lambda(\rho_n(z)1). \end{align*} Since $\rho_n(z)1=z^n\cdot 1=z^n$, applying $\lambda$ gives \begin{align*} c_{\lambda,1}(z)=\lambda(z^n)=z^n. \end{align*} Thus the matrix coefficients obtained from these irreducible representations are exactly the usual Fourier modes $z^n$, and their finite linear combinations are Laurent polynomials $\sum_{n=-N}^N a_n z^n$ on $\mathbb T$. This is the abelian boundary case: because every irreducible representation of $\mathbb T$ is one-dimensional, each matrix coefficient is already the trace of the representation up to multiplication by the chosen vector and functional, so matrix coefficients and ordinary characters carry the same Fourier data. [/example] The example shows what the Peter-Weyl theorem should generalize: Fourier modes become matrix coefficients. To make orthogonality statements invariant under a [change of basis](/page/Change%20Of%20Basis), we introduce unitary matrix coefficients, which use the averaged invariant inner product available on compact groups. We normalize Haar measure $dg$ by $\int_G 1\,dg=1$. [definition: Unitary Matrix Coefficient] Let $(\rho,V)$ be a finite-dimensional unitary representation of a compact Lie group $G$, where $V$ has Hermitian inner product $(\cdot,\cdot)_V$ linear in the first argument. For $v,w\in V$, the associated unitary matrix coefficient is the function $c_{w,v}:G\to\mathbb C$ defined by \begin{align*} c_{w,v}(g)=(\rho(g)v,w)_V. \end{align*} [/definition] Unitary coefficients are better suited to integration because adjoints behave well under inversion in $G$. The obstacle is that matrix coefficients from different representations are just functions on the same group, so without an invariant integral identity there is no reason for them to separate into orthogonal blocks. To build Fourier theory on $G$, we need a precise rule saying when two such coefficients are orthogonal and what their norm is. Irreducibility should force any averaged intertwining operator to collapse to a scalar or to zero, and that scalar is what determines the normalization of the orthogonality relation. [quotetheorem:9715] [citeproof:9715] Schur orthogonality is the compact-group analogue of the orthogonality of exponential functions on the circle, but its hypotheses are doing real work. Irreducibility is what lets Schur's lemma collapse the averaged operator to either $0$ or a scalar multiple of the identity; for reducible representations the same integral decomposes block by block rather than giving a single scalar formula. Unitarity, obtained by averaging an inner product over compact $G$, makes complex conjugation match adjoints and inversion, while normalized Haar measure fixes the constant $1/\dim V$ rather than leaving an arbitrary volume factor. Compactness is essential here: for a noncompact group such as $\mathbb R$, ordinary matrix coefficients need not be square-integrable against Haar measure, and the discrete orthogonality statement is replaced by continuous Fourier analysis. The theorem also does not yet say that all functions on $G$ have such expansions; it only says that the irreducible coefficient functions, once they appear, are mutually orthogonal with the displayed normalization. [example: Regular Representation of a Finite Group] Let $G$ be finite and give it normalized counting measure, so integration means \begin{align*} \int_G F(g)\,dg=\frac{1}{|G|}\sum_{g\in G}F(g). \end{align*} The left regular representation acts on $\mathbb C[G]=\{f:G\to\mathbb C\}$ by \begin{align*} (L_hf)(g)=f(h^{-1}g). \end{align*} Since a function on $G$ is determined by its values at the $|G|$ elements of $G$, the delta functions $\delta_x(g)=1$ if $g=x$ and $\delta_x(g)=0$ otherwise form a basis of $\mathbb C[G]$, so \begin{align*} \dim \mathbb C[G]=|G|. \end{align*} For a finite group, Schur orthogonality identifies the regular representation, with its commuting left and right translation actions, as \begin{align*} \mathbb C[G]\cong \bigoplus_{\pi\in\widehat G}V_\pi\otimes V_\pi^* \end{align*} as a $G\times G$-representation. The dimension of the tensor factor indexed by $\pi$ is \begin{align*} \dim(V_\pi\otimes V_\pi^*)=(\dim V_\pi)(\dim V_\pi^*). \end{align*} Because $V_\pi^*$ has the [dual basis](/theorems/414) to any basis of $V_\pi$, we have \begin{align*} \dim V_\pi^*=\dim V_\pi. \end{align*} Therefore \begin{align*} \dim(V_\pi\otimes V_\pi^*)=(\dim V_\pi)^2. \end{align*} Taking dimensions in the displayed direct sum gives \begin{align*} |G|=\dim\mathbb C[G]=\sum_{\pi\in\widehat G}\dim(V_\pi\otimes V_\pi^*)=\sum_{\pi\in\widehat G}(\dim V_\pi)^2. \end{align*} Thus the regular representation splits into one matrix block $V_\pi\otimes V_\pi^*$ for each irreducible representation $\pi$, and the identity $|G|=\sum_{\pi\in\widehat G}(\dim V_\pi)^2$ is the finite-dimensional shadow of the Peter-Weyl decomposition. [/example] The finite group comparison is useful because it contains no analytic closure issue. For compact Lie groups, the same decomposition appears inside an infinite-dimensional Hilbert space, so density must be proved. ## The Peter-Weyl Decomposition of $L^2(G)$ Which functions on $G$ can be reconstructed from finite-dimensional representations? The Peter-Weyl theorem says that the representation-theoretic coefficients are not merely many examples: they span a dense subspace of the Hilbert space $L^2(G)$. [definition: Representative Functions] Let $G$ be a compact Lie group. The space of representative functions, denoted $\mathcal R(G)$, is the complex linear span in $C(G)$ of all matrix coefficients of all finite-dimensional continuous complex representations of $G$. [/definition] The word representative records that these functions are represented by finite-dimensional linear data. Before proving density, we need to know that these functions form an algebra robust enough for Stone-Weierstrass: products, conjugates, constants, and point separation must remain inside the same class. [quotetheorem:9741] [citeproof:9741] Each condition in this algebra theorem is precisely one of the Stone-Weierstrass requirements. Products force finite-dimensionality to be stable under tensor products, conjugates require passing to dual or conjugate representations, constants come from the trivial representation, and point separation prevents the algebra from seeing only a quotient of $G$. If point separation failed, Stone-Weierstrass would at best approximate functions constant on the indistinguishable fibres, not arbitrary functions on $G$. Continuity is also essential because the density step first occurs in $C(G)$ with the [uniform norm](/page/Uniform%20Norm); discontinuous coefficients would not belong to that Banach algebra. Compactness supplies both the uniform approximation theorem on $G$ and the Haar probability measure used later to pass from $C(G)$ to $L^2(G)$. The remaining question is completeness in the Hilbert space sense: once Schur orthogonality gives an orthogonal family, does this family exhaust $L^2(G)$? [quotetheorem:8833] [citeproof:8833] The theorem is the non-abelian Fourier expansion theorem for compact groups. A function $f\in L^2(G)$ has Fourier components in the finite-dimensional blocks $V_\pi\otimes V_\pi^*$ rather than scalar Fourier coefficients alone. The Hilbert direct sum completion is necessary because a general $L^2$ function is usually an infinite series of coefficient functions, just as an arbitrary square-integrable function on $\mathbb T$ need not be a trigonometric polynomial. Compactness is again decisive: it gives a discrete unitary dual in this theorem and finite Haar measure, whereas noncompact groups such as $\mathbb R$ have continuous spectral decompositions rather than a countable orthonormal basis of finite-dimensional matrix coefficients. The statement is also an $L^2$ theorem, so it gives convergence in norm and not automatic pointwise or [uniform convergence](/page/Uniform%20Convergence) for every function. [example: Peter-Weyl for $\mathbb T$] For $G=\mathbb T$, every irreducible representation is one-dimensional and has the form $\rho_n(z)w=z^n w$ for some $n\in\mathbb Z$. Since $\dim \rho_n=1$, the normalization factor in the *Peter-Weyl Theorem* is $\sqrt{\dim \rho_n}=1$, and the unique matrix coefficient of $\rho_n$ is the function $z\mapsto z^n$. With normalized Haar measure, write $z=e^{i\theta}$ and integrate by \begin{align*} \int_{\mathbb T} f(z)\,dz=\frac{1}{2\pi}\int_0^{2\pi} f(e^{i\theta})\,d\theta. \end{align*} For $m,n\in\mathbb Z$, the $L^2$ inner product of the corresponding coefficients is \begin{align*} \langle z^n,z^m\rangle=\frac{1}{2\pi}\int_0^{2\pi} e^{in\theta}\overline{e^{im\theta}}\,d\theta. \end{align*} Since $\overline{e^{im\theta}}=e^{-im\theta}$, this becomes \begin{align*} \langle z^n,z^m\rangle=\frac{1}{2\pi}\int_0^{2\pi} e^{i(n-m)\theta}\,d\theta. \end{align*} If $n=m$, the integrand is $1$, so \begin{align*} \langle z^n,z^n\rangle=\frac{1}{2\pi}\int_0^{2\pi}1\,d\theta=1. \end{align*} If $n\ne m$, then \begin{align*} \int_0^{2\pi} e^{i(n-m)\theta}\,d\theta=\frac{e^{i(n-m)2\pi}-1}{i(n-m)}. \end{align*} Because $n-m\in\mathbb Z$, we have $e^{i(n-m)2\pi}=1$, hence \begin{align*} \int_0^{2\pi} e^{i(n-m)\theta}\,d\theta=0. \end{align*} Thus $\langle z^n,z^m\rangle=\delta_{nm}$, and Peter-Weyl says that these orthonormal functions span a dense subspace of $L^2(\mathbb T)$. Therefore $\{z^n:n\in\mathbb Z\}$ is an orthonormal basis of $L^2(\mathbb T)$, which is exactly the Hilbert-space form of ordinary Fourier series. [/example] The abelian case has no matrix indices because every irreducible representation is one-dimensional. For non-abelian groups, the matrix block attached to an irreducible representation is a genuine matrix algebra of coefficients. [example: The First $SU(2)$ Matrix Blocks] Let $G=SU(2)$ and let $V_m=\operatorname{Sym}^m(\mathbb C^2)$ be the irreducible representation of highest weight $m$. If $e_1,e_2$ is the standard basis of $\mathbb C^2$, then the monomials $e_1^{m-r}e_2^r$ for $0\le r\le m$ form a basis of $V_m$, so \begin{align*} \dim V_m=m+1. \end{align*} By the *Peter-Weyl Theorem*, the block attached to $V_m$ in $L^2(SU(2))$ is the coefficient space $V_m\otimes V_m^*$. Since $\dim V_m^*=\dim V_m$, its dimension is \begin{align*} \dim(V_m\otimes V_m^*)=(\dim V_m)(\dim V_m^*)=(m+1)(m+1)=(m+1)^2. \end{align*} For $m=0$, we have $V_0=\operatorname{Sym}^0(\mathbb C^2)\cong\mathbb C$. The action is trivial, so $g\cdot 1=1$ for every $g\in SU(2)$. If $v=\alpha\in\mathbb C$ and $\lambda\in V_0^*$, then the corresponding coefficient is \begin{align*} c_{\lambda,\alpha}(g)=\lambda(g\cdot \alpha)=\lambda(\alpha). \end{align*} This is independent of $g$, hence the $m=0$ block is exactly the one-dimensional space of constant functions. For $m=1$, we have $V_1=\mathbb C^2$, the defining representation. Write the entries of $g\in SU(2)$ in the standard form determined by complex numbers $a,b$ with $|a|^2+|b|^2=1$, so that \begin{align*} g e_1=a e_1-\overline b e_2. \end{align*} Also, \begin{align*} g e_2=b e_1+\overline a e_2. \end{align*} Using the Hermitian inner product for which $e_1,e_2$ is orthonormal and which is linear in the first argument, the four matrix coefficients are \begin{align*} (g e_1,e_1)=a. \end{align*} Similarly, \begin{align*} (g e_2,e_1)=b. \end{align*} The remaining two are \begin{align*} (g e_1,e_2)=-\overline b. \end{align*} and \begin{align*} (g e_2,e_2)=\overline a. \end{align*} Thus the $m=1$ block is the four-dimensional span of the coefficient functions $a$, $b$, $-\overline b$, and $\overline a$. For general $m$, the action on a basis monomial is found by substituting the two displayed formulas for $g e_1$ and $g e_2$: \begin{align*} g(e_1^{m-r}e_2^r)=(a e_1-\overline b e_2)^{m-r}(b e_1+\overline a e_2)^r. \end{align*} Expanding the first factor by the [binomial theorem](/theorems/750) gives terms indexed by $0\le p\le m-r$ of the form \begin{align*} \binom{m-r}{p}a^{m-r-p}(-\overline b)^p e_1^{m-r-p}e_2^p. \end{align*} Expanding the second factor gives terms indexed by $0\le q\le r$ of the form \begin{align*} \binom{r}{q}b^{r-q}\overline a^q e_1^{r-q}e_2^q. \end{align*} Multiplying one term from the first expansion and one term from the second gives the coefficient factor \begin{align*} \binom{m-r}{p}\binom{r}{q}a^{m-r-p}(-\overline b)^p b^{r-q}\overline a^q. \end{align*} The total degree in $a$, $b$, $\overline a$, and $\overline b$ is \begin{align*} (m-r-p)+p+(r-q)+q=m. \end{align*} Therefore every coefficient in the $m$th block is a polynomial of total degree $m$ in the entries $a$, $b$, $\overline a$, and $\overline b$. The first Peter-Weyl blocks for $SU(2)$ are constants for $m=0$, the defining matrix entries for $m=1$, and higher polynomial matrix coefficients obtained from the symmetric powers. [/example] This block decomposition is stronger than a statement about characters. Characters only see the trace inside each block, but class functions are precisely the part of $L^2(G)$ where traces become sufficient. ## Characters and Class Functions What changes when a function is constant on conjugacy classes? Such a function cannot distinguish individual matrix entries inside an irreducible representation; conjugation invariance collapses each matrix block to a one-dimensional trace direction. [definition: Character] Let $(\rho,V)$ be a finite-dimensional complex representation of a compact Lie group $G$. Its character is the function $\chi_\rho:G\to\mathbb C$ defined by \begin{align*} \chi_\rho(g)=\operatorname{Tr}(\rho(g)). \end{align*} [/definition] Characters are class functions because trace is invariant under conjugation. To state the Hilbert-space theorem they satisfy, we isolate the class functions as the conjugation-invariant part of the function space. [definition: Class Function] Let $G$ be a compact Lie group. A function $f:G\to\mathbb C$ is a class function if \begin{align*} f(hgh^{-1})=f(g) \end{align*} for all $g,h\in G$. [/definition] The Hilbert space of square-integrable class functions is the closed subspace of $L^2(G)$ fixed by the conjugation action. Schur orthogonality already gives orthogonality of irreducible characters, but orthogonality alone does not prove that no conjugation-invariant functions are missing from their span. The remaining problem is completeness: every square-integrable class function should be recoverable from its character Fourier coefficients. [quotetheorem:9742] [citeproof:9742] Character orthogonality is the Fourier basis theorem after imposing conjugation symmetry. Irreducibility is necessary for the inner product to be exactly $\delta_{\pi\sigma}$; the character of a reducible representation is a sum of irreducible characters, so its norm records multiplicities rather than being automatically $1$. Compactness and unitarity enter through Schur orthogonality and the normalized Haar integral. The class-function restriction is also essential: for a non-abelian group, characters do not span all of $L^2(G)$ because they only see the trace direction inside each matrix block and discard the off-diagonal coefficient functions. Thus character theory is complete for conjugation-invariant harmonic analysis, not for arbitrary functions on $G$. [example: $SU(2)$ Characters] Every element of $SU(2)$ is conjugate to a diagonal matrix of the form \begin{align*} t_\theta=\operatorname{diag}(e^{i\theta},e^{-i\theta}). \end{align*} For $V_m=\operatorname{Sym}^m(\mathbb C^2)$, use the basis \begin{align*} e_1^m,\ e_1^{m-1}e_2,\ \ldots,\ e_2^m. \end{align*} On the basis vector $e_1^{m-r}e_2^r$, the diagonal matrix $t_\theta$ acts by \begin{align*} t_\theta(e_1^{m-r}e_2^r)=(e^{i\theta}e_1)^{m-r}(e^{-i\theta}e_2)^r. \end{align*} Multiplying the scalar factors gives \begin{align*} (e^{i\theta})^{m-r}(e^{-i\theta})^r=e^{i(m-r)\theta}e^{-ir\theta}=e^{i(m-2r)\theta}. \end{align*} Thus the diagonal entries of $t_\theta$ on this basis are $e^{i(m-2r)\theta}$ for $0\le r\le m$, so the character is \begin{align*} \chi_m(\theta)=\sum_{r=0}^m e^{i(m-2r)\theta}=e^{im\theta}+e^{i(m-2)\theta}+\cdots+e^{-im\theta}. \end{align*} If $\sin\theta\ne 0$, this finite geometric series has first term $e^{im\theta}$ and ratio $e^{-2i\theta}$, hence \begin{align*} \chi_m(\theta)=e^{im\theta}\frac{1-e^{-2i(m+1)\theta}}{1-e^{-2i\theta}}. \end{align*} Rewrite numerator and denominator as \begin{align*} 1-e^{-2i(m+1)\theta}=e^{-i(m+1)\theta}(e^{i(m+1)\theta}-e^{-i(m+1)\theta}). \end{align*} Also, \begin{align*} 1-e^{-2i\theta}=e^{-i\theta}(e^{i\theta}-e^{-i\theta}). \end{align*} Substituting these identities gives \begin{align*} \chi_m(\theta)=e^{im\theta}\frac{e^{-i(m+1)\theta}(e^{i(m+1)\theta}-e^{-i(m+1)\theta})}{e^{-i\theta}(e^{i\theta}-e^{-i\theta})}. \end{align*} The exponential prefactor is \begin{align*} e^{im\theta}e^{-i(m+1)\theta}e^{i\theta}=1. \end{align*} Using $e^{ix}-e^{-ix}=2i\sin x$, we obtain \begin{align*} \chi_m(\theta)=\frac{2i\sin((m+1)\theta)}{2i\sin\theta}=\frac{\sin((m+1)\theta)}{\sin\theta}. \end{align*} When $\theta\in\pi\mathbb Z$, the character is defined by the original finite sum; if $\theta=k\pi$, then each term equals $(-1)^{mk}$, so \begin{align*} \chi_m(k\pi)=\sum_{r=0}^m (-1)^{mk}=(m+1)(-1)^{mk}. \end{align*} Finally, \begin{align*} \chi_m(-\theta)=\frac{\sin(-(m+1)\theta)}{\sin(-\theta)}=\frac{-\sin((m+1)\theta)}{-\sin\theta}=\chi_m(\theta) \end{align*} away from $\pi\mathbb Z$, and the finite-sum formula gives the same equality at $\theta\in\pi\mathbb Z$. Thus the $SU(2)$ characters are trigonometric polynomials on the maximal torus and are invariant under the Weyl group action $\theta\mapsto-\theta$. [/example] This example previews the Weyl character formula: in rank one, class functions become Weyl-invariant functions of a single torus parameter. The separation theorem below explains why knowing all irreducible characters is not only enough for $L^2$ approximation, but also enough to distinguish actual conjugacy classes. [quotetheorem:9743] [citeproof:9743] This result is the conceptual reason characters are effective coordinates on the set of conjugacy classes. The conclusion depends on compactness in two ways: conjugacy classes are compact enough to be separated by continuous class functions, and Peter-Weyl supplies enough finite-dimensional characters to approximate those functions. Without the class-function viewpoint, characters cannot distinguish elements inside the same conjugacy class, since they are constant there by definition; for example, two non-equal conjugate matrices in $SU(2)$ have identical values under every character. The theorem therefore says exactly that this is the only ambiguity. Later, after choosing a maximal torus $T$, the theorem combines with conjugacy into $T$ and the Weyl group action to identify class functions on $G$ with Weyl-invariant functions on $T$. [example: Separation on $SU(2)$] Every element of $SU(2)$ is conjugate to some \begin{align*} t_\theta=\operatorname{diag}(e^{i\theta},e^{-i\theta}), \end{align*} and replacing $\theta$ by $-\theta$ only swaps the two eigenvalues. Thus $t_\theta$ and $t_\phi$ are conjugate exactly when $\phi\equiv\theta \pmod {2\pi}$ or $\phi\equiv-\theta \pmod {2\pi}$. For the defining representation on $\mathbb C^2$, the character is the trace: \begin{align*} \chi_1(t_\theta)=e^{i\theta}+e^{-i\theta}=2\cos\theta. \end{align*} If $\chi_1(t_\theta)=\chi_1(t_\phi)$, then $\cos\theta=\cos\phi$. Using \begin{align*} \cos\theta-\cos\phi=-2\sin\left(\frac{\theta+\phi}{2}\right)\sin\left(\frac{\theta-\phi}{2}\right), \end{align*} we get either $\sin((\theta+\phi)/2)=0$ or $\sin((\theta-\phi)/2)=0$. Hence $\theta+\phi\in 2\pi\mathbb Z$ or $\theta-\phi\in 2\pi\mathbb Z$, so $\phi\equiv-\theta$ or $\phi\equiv\theta$ modulo $2\pi$. Therefore the single character $\chi_1$ already separates conjugacy classes in $SU(2)$. The higher characters are also functions of the same quantity $2\cos\theta$. From the previous computation, \begin{align*} \chi_m(\theta)=e^{im\theta}+e^{i(m-2)\theta}+\cdots+e^{-im\theta}. \end{align*} Let $x=2\cos\theta=e^{i\theta}+e^{-i\theta}$. Then $\chi_0=1$ and $\chi_1=x$. Multiplying the finite sum for $\chi_m$ by $x$ shifts it once upward and once downward: \begin{align*} x\chi_m(\theta)=(e^{i\theta}+e^{-i\theta})(e^{im\theta}+e^{i(m-2)\theta}+\cdots+e^{-im\theta}). \end{align*} The upward-shifted terms are $e^{i(m+1)\theta},e^{i(m-1)\theta},\ldots,e^{-i(m-1)\theta}$, and the downward-shifted terms are $e^{i(m-1)\theta},e^{i(m-3)\theta},\ldots,e^{-i(m+1)\theta}$. The two endpoint terms occur once, while the middle terms occur twice, so subtracting \begin{align*} \chi_{m-1}(\theta)=e^{i(m-1)\theta}+e^{i(m-3)\theta}+\cdots+e^{-i(m-1)\theta} \end{align*} leaves \begin{align*} \chi_{m+1}(\theta)=x\chi_m(\theta)-\chi_{m-1}(\theta). \end{align*} By induction, each $\chi_m$ is a polynomial in $x=2\cos\theta=\chi_1(\theta)$. Thus all irreducible $SU(2)$ characters are generated by the defining character, and the defining character already contains exactly the conjugacy-class parameter. [/example] ## From Peter-Weyl to Weyl's Formulae What has this chapter added to the structure theory of compact Lie groups? It supplies the analytic decomposition that lets representation theory be read from functions. Matrix coefficients decompose the whole Hilbert space $L^2(G)$, and irreducible characters decompose the class-function part. [remark: Fourier Analysis Dictionary] For $G=\mathbb T$, the Fourier modes $z^n$ are both matrix coefficients and characters. For a non-abelian compact group, matrix coefficients play the role of all Fourier modes, while characters play the role of the conjugation-invariant modes. The extra matrix indices record the failure of irreducible representations to be one-dimensional. [/remark] The next step in the course is to compute inner products of class functions by reducing integration over $G$ to integration over a maximal torus. Peter-Weyl explains why this is enough: once the character inner products are computable, the irreducible characters become an explicit orthonormal basis. Peter-Weyl shows that irreducible representations are not just algebraic objects but the building blocks of harmonic analysis on `G`. To compute their inner products and exploit class-function symmetry, we next reduce integration on the whole group to integration over a maximal torus using the Weyl integration formula. # 8. Weyl Integration Formula Chapters 2 and 3 showed that a compact connected Lie group $G$ is controlled, up to conjugacy, by a maximal torus $T$ and its Weyl group $W=N_G(T)/T$; Chapter 4 supplied the roots that now determine the Jacobian. This chapter turns that structural statement into an integration theorem. The question is how Haar integration on $G$ can be reduced to integration over $T$, and why the reduction must include both a finite symmetry factor $|W|$ and a root-theoretic Jacobian. The main geometric object is the conjugation map from $G/T \times T$ to $G$. Its regular part has finite fibres indexed by the Weyl group, while its differential contributes the Weyl denominator. The resulting Weyl integration formula is the analytic form of the principle that class functions on $G$ live on $T/W$. ## The Conjugation Map and Regular Elements The first problem is to understand how many times a typical element of $G$ is represented as a conjugate of an element of $T$. The maximal torus theorem gives surjectivity of conjugation from $G \times T$, but integration requires a local calculation with the correct quotient by the redundant right action of $T$. Let $G$ be a compact connected Lie group, let $T\subset G$ be a maximal torus, and write $\mathfrak g$ and $\mathfrak t$ for their Lie algebras. The conjugation map descends to $G/T$ because elements of $T$ commute with $T$. [definition: Conjugation Map] The conjugation map associated to $T$ is the smooth map $\Phi:G/T\times T\to G$ defined by \begin{align*} \Phi(gT,t)=gtg^{-1}. \end{align*} [/definition] This map is surjective by the [conjugacy theorem for maximal tori](/theorems/9720), but its fibres still contain a finite redundancy coming from elements that move $T$ to itself. To name this finite ambiguity before counting fibres, we need the Weyl group action on the torus. [definition: Weyl Group Action on the Torus] The Weyl group action on the torus is the map \begin{align*} W\times T&\to T \end{align*} given by \begin{align*} (nT,t)&\mapsto ntn^{-1}. \end{align*} [/definition] The quotient $T/W$ should be thought of as the space of conjugacy classes in $G$. For the conjugation map to behave like a finite covering, we must isolate exactly those torus elements whose centraliser has no directions outside $T$; this motivates the definition of regular elements. [definition: Regular Element of the Torus] An element $t\in T$ is regular if its centraliser in $G$ is exactly $T$: \begin{align*} C_G(t)=T. \end{align*} The set of regular elements in $T$ is denoted $T_{\mathrm{reg}}$. [/definition] Regularity says that $t$ has no additional symmetry beyond the torus. This is precisely the hypothesis needed for the Weyl group to account for all remaining multiplicity in the conjugation parametrisation. [quotetheorem:9744] [citeproof:9744] This result isolates the only global overcounting on the regular set: the Weyl group. Regularity is essential here. In $SU(2)$, the central elements $I$ and $-I$ are fixed by every conjugation, so the map $G/T\times T\to G$ cannot look locally like a $|W|$-sheeted covering above them; the entire $G/T$ direction collapses. Thus the theorem does not describe the singular fibres, nor does it say that every element of $G$ has exactly $|W|$ representatives in $G/T\times T$. It says that after removing the singular locus, the only remaining finite ambiguity is Weyl symmetry. The next issue is local and analytic: on this regular covering, how much volume distortion occurs when the coordinates $(gT,t)$ are pushed forward to $G$ by conjugation? [example: Regular Elements in SU(2)] Let $G=SU(2)$ and write \begin{align*} t_\theta=\operatorname{diag}(e^{i\theta},e^{-i\theta}). \end{align*} Every element $u\in SU(2)$ has entries $u_{11}=a$, $u_{12}=b$, $u_{21}=-\overline b$, and $u_{22}=\overline a$, with $|a|^2+|b|^2=1$. Put $\lambda=e^{i\theta}$. Comparing the $(1,2)$-entries of $ut_\theta$ and $t_\theta u$ gives \begin{align*} b\lambda^{-1}=\lambda b. \end{align*} Thus \begin{align*} b(\lambda^{-1}-\lambda)=0. \end{align*} The $(2,1)$-entries give the conjugate condition \begin{align*} -\overline b\lambda=-\lambda^{-1}\overline b. \end{align*} If $\lambda^2\ne 1$, then $\lambda^{-1}\ne\lambda$, so $b=0$. Hence $u=\operatorname{diag}(a,\overline a)$, and since $|a|=1$ this element lies in $T$. Therefore $C_G(t_\theta)=T$ whenever $e^{2i\theta}\ne 1$. If $\lambda^2=1$, then $\lambda=\pm 1$, so $t_\theta=\pm I$. These matrices commute with every element of $SU(2)$, hence \begin{align*} C_G(t_\theta)=SU(2) \end{align*} for $\theta=0$ or $\theta=\pi$ modulo $2\pi$. Therefore \begin{align*} T_{\mathrm{reg}}=T\setminus\{I,-I\}. \end{align*} The normalizer has one nontrivial coset represented by the element $n\in SU(2)$ with entries $n_{11}=0$, $n_{12}=-1$, $n_{21}=1$, and $n_{22}=0$. Since $n^{-1}=-n$, conjugating gives \begin{align*} nt_\theta n^{-1}=\operatorname{diag}(e^{-i\theta},e^{i\theta})=t_{-\theta}. \end{align*} Also $n^2=-I\in T$, so $nT$ has order $2$ in $N_G(T)/T$. Thus $W\cong \mathbb Z/2\mathbb Z$, and its nontrivial element acts on $T$ by $\theta\mapsto -\theta$. [/example] The example shows why singular elements have measure zero but cannot be ignored conceptually: they are precisely where the finite covering picture degenerates. ## The Weyl Denominator as a Jacobian The local calculation asks for the determinant of the differential of $\Phi$ in directions transverse to the torus. Since $T$ itself commutes with $t$, the relevant tangent directions are the quotient $\mathfrak g/\mathfrak t$, and roots diagonalise the adjoint action of $T$ on this quotient. Fix an invariant inner product on $\mathfrak g$. Let $R\subset i\mathfrak t^*$ be the root system of $(G,T)$ after complexifying $\mathfrak g$, and choose a set $R^+$ of positive roots. The root decomposition is \begin{align*} \mathfrak g_{\mathbb C}=\mathfrak t_{\mathbb C}\oplus \bigoplus_{\alpha\in R}\mathfrak g_\alpha, \end{align*} and $t\in T$ acts on $\mathfrak g_\alpha$ by the character $e^{\alpha}(t)$. [definition: Weyl Denominator Density] The Weyl denominator density is the function $D:T\to \mathbb R_{\ge 0}$ defined by \begin{align*} D(t)=\prod_{\alpha\in R^+}\left|1-e^\alpha(t)\right|^2. \end{align*} [/definition] This is the globally defined object on $T$ needed for integration. In a simply connected semisimple setting, or after pulling back to the Lie algebra or the universal cover of $T$, one may choose half-root factors and write a local alternating product \begin{align*} \Delta=\prod_{\alpha\in R^+}(e^{\alpha/2}-e^{-\alpha/2}), \end{align*} but those half-root characters need not descend to honest characters on $T$ for an arbitrary compact connected Lie group. The density $D=|\Delta|^2$ is independent of this ambiguity and is $W$-invariant. We now need to identify this invariant density with the Jacobian from the conjugation coordinates. [quotetheorem:9746] [citeproof:9746] The formula also explains the singular locus: $t$ fails to be regular precisely when some root evaluates to $1$ on $t$. Thus the Jacobian vanishes on the root hypertori, the compact analogue of the discriminant locus. Dropping regularity would make the displayed determinant misleading as a change-of-variables factor, because the differential is no longer invertible in the transverse directions. In $SU(2)$ at $t=I$ or $t=-I$, the conjugation map is constant along the $G/T$ coordinate, so its transverse Jacobian is $0$ rather than the Jacobian of a local coordinate chart. The theorem therefore computes the Jacobian only on the regular locus; the singular set contributes no Haar measure in the final integral, but it is exactly where the covering and local diffeomorphism picture breaks down. This vanishing is what will make the Weyl density suppress the singular conjugacy classes in the integration formula. The displayed normalization also depends on the measure convention. The invariant inner product fixes compatible Riemannian measures on $G$, $T$, and the quotient $G/T$; replacing these by rescaled Haar measures rescales the Jacobian statement by the corresponding quotient-volume factor. The later normalized Weyl formula absorbs this convention into probability Haar measures, leaving the density $D(t)$ and the finite factor $|W|$ as the intrinsic pieces. [example: The SU(2) Jacobian] For $SU(2)$ the root system has one positive root $\alpha$, and for \begin{align*} t_\theta=\operatorname{diag}(e^{i\theta},e^{-i\theta}) \end{align*} the corresponding root character is \begin{align*} e^\alpha(t_\theta)=e^{i\theta}(e^{-i\theta})^{-1}=e^{2i\theta}. \end{align*} Therefore the Weyl denominator density is \begin{align*} D(t_\theta)=|1-e^{2i\theta}|^2. \end{align*} Using $e^{2i\theta}=\cos(2\theta)+i\sin(2\theta)$, we get \begin{align*} |1-e^{2i\theta}|^2=|1-\cos(2\theta)-i\sin(2\theta)|^2. \end{align*} Thus \begin{align*} |1-e^{2i\theta}|^2=(1-\cos(2\theta))^2+\sin^2(2\theta). \end{align*} Expanding and using $\cos^2(2\theta)+\sin^2(2\theta)=1$ gives \begin{align*} (1-\cos(2\theta))^2+\sin^2(2\theta)=2-2\cos(2\theta). \end{align*} Since $1-\cos(2\theta)=2\sin^2\theta$, this becomes \begin{align*} D(t_\theta)=4\sin^2\theta. \end{align*} The density vanishes exactly when $\sin\theta=0$, equivalently $\theta=0$ or $\theta=\pi$ modulo $2\pi$; these are precisely the central elements $I$ and $-I$, where regularity fails. [/example] The determinant has now supplied the continuous weight, while the covering theorem supplied the discrete factor. Combining these two ingredients gives the integration formula. ## Integration of Class Functions The motivating problem is to integrate a function on $G$ that is constant on conjugacy classes. Since such a function is determined by its restriction to $T$, Weyl integration gives the precise rule for replacing the integral over $G$ by an integral over $T$. [definition: Class Function] A [measurable function](/page/Measurable%20Function) $f:G\to \mathbb C$ is a class function if \begin{align*} f(hgh^{-1})=f(g) \end{align*} for all $g,h\in G$. [/definition] Class functions remove the dependence on the $G/T$ coordinate in the conjugation map. The previous covering and Jacobian results now combine into the central formula of the chapter. [quotetheorem:9714] [citeproof:9714] This is the central analytic statement of the chapter. It says that the quotient $T/W$ is not integrated with flat torus measure; it carries the density $D$, which records the size of conjugacy classes. Each hypothesis has a role. Compactness gives finite normalized Haar measure; for a noncompact group such as $\mathbb R$ there is no probability Haar measure of this kind. Connectedness ensures that every element is conjugate into a maximal torus; in a disconnected group such as $O(2)$, reflections are not conjugate into the rotation torus $SO(2)$. Maximality of $T$ is also necessary: the subgroup $\{I\}\subset SU(2)$ cannot parametrize noncentral conjugacy classes. Finally, the class-function hypothesis is essential because restriction to $T$ forgets the $G/T$ directions; a continuous function supported in a small [open set](/page/Open%20Set) disjoint from $T$ has zero restriction to $T$ but can have positive integral over $G$. Thus the theorem does not reconstruct or integrate arbitrary functions from their values on $T$; it is an integration theorem for conjugation-invariant functions. [remark: Normalization of the Weyl Formula] Some texts use unnormalised Riemannian volume on $G$, $T$, and $G/T$. In that convention the formula contains an extra factor involving $\operatorname{vol}(G/T)$, and the displayed normalized formula is recovered after rescaling Haar measures to total mass $1$. The factor $1/|W|$ remains in all conventions because it comes from the finite covering degree. [/remark] This repeats the boundary of the theorem in operational form: restriction to $T$ is an integration device only after conjugation invariance has removed the $G/T$ directions. [example: SU(2) Class Integrals] Let $f:SU(2)\to\mathbb C$ be a class function and write $f(t_\theta)=F(\theta)$ for $t_\theta=\operatorname{diag}(e^{i\theta},e^{-i\theta})$. The nontrivial Weyl element sends $t_\theta$ to $t_{-\theta}$, so $F(-\theta)=F(\theta)$, and $F$ is $2\pi$-periodic because $t_{\theta+2\pi}=t_\theta$. For $SU(2)$ we have $|W|=2$ and, from the $SU(2)$ Jacobian computation, $D(t_\theta)=4\sin^2\theta$. With normalized Haar measure $dt=d\theta/(2\pi)$ on $T\cong S^1$, the *Weyl Integration Formula* gives \begin{align*} \int_{SU(2)} f(g)\,dg=\frac{1}{2}\int_0^{2\pi}F(\theta)\,4\sin^2\theta\,\frac{d\theta}{2\pi}. \end{align*} The scalar factor is \begin{align*} \frac{1}{2}\cdot 4\cdot \frac{1}{2\pi}=\frac{1}{\pi}. \end{align*} Hence \begin{align*} \int_{SU(2)} f(g)\,dg=\frac{1}{\pi}\int_0^{2\pi}F(\theta)\sin^2\theta\,d\theta. \end{align*} Since $F(-\theta)=F(\theta)$ and $\sin^2(-\theta)=\sin^2\theta$, the product $F(\theta)\sin^2\theta$ is even. Its $2\pi$-periodicity gives \begin{align*} \int_0^{2\pi}F(\theta)\sin^2\theta\,d\theta=\int_{-\pi}^{\pi}F(\theta)\sin^2\theta\,d\theta. \end{align*} Evenness then gives \begin{align*} \int_{-\pi}^{\pi}F(\theta)\sin^2\theta\,d\theta=2\int_0^\pi F(\theta)\sin^2\theta\,d\theta. \end{align*} Therefore the same integral may be written over the Weyl chamber $0\le \theta\le \pi$ as \begin{align*} \int_{SU(2)} f(g)\,dg=\frac{2}{\pi}\int_0^\pi F(\theta)\sin^2\theta\,d\theta. \end{align*} The endpoints $\theta=0$ and $\theta=\pi$ are the singular central elements, but they do not change the integral because they form a measure-zero subset of the chamber. [/example] The same mechanism becomes the eigenvalue integration formula for unitary groups. The roots encode pairwise collisions of eigenvalues, and the Weyl denominator becomes the Vandermonde determinant. [example: Eigenvalue Integration for $U(n)$] Let $G=U(n)$ and let $T$ be the diagonal unitary subgroup, so an element of $T$ has the form \begin{align*} t=\operatorname{diag}(z_1,\dots,z_n),\qquad |z_j|=1. \end{align*} Permutation matrices normalize $T$ by permuting the diagonal entries, and quotienting by the diagonal subgroup leaves exactly the permutation ambiguity. Thus the Weyl group is $S_n$, so $|W|=n!$. For the positive roots indexed by pairs $1\le i<j\le n$, the root character is \begin{align*} e^{\alpha_{ij}}(t)=z_i z_j^{-1}. \end{align*} Therefore the Weyl denominator density is \begin{align*} D(t)=\prod_{1\le i<j\le n}|1-z_i z_j^{-1}|^2. \end{align*} Since $|z_j|=1$, multiplication by $z_j$ preserves absolute value, and \begin{align*} |1-z_i z_j^{-1}|^2=|z_j(1-z_i z_j^{-1})|^2. \end{align*} The factor inside the absolute value is \begin{align*} z_j(1-z_i z_j^{-1})=z_j-z_i. \end{align*} Hence \begin{align*} |1-z_i z_j^{-1}|^2=|z_j-z_i|^2=|z_i-z_j|^2. \end{align*} Substituting this identity into the product gives \begin{align*} D(t)=\prod_{1\le i<j\le n}|z_i-z_j|^2. \end{align*} If $f:U(n)\to\mathbb C$ is a class function and \begin{align*} F(z_1,\dots,z_n)=f(\operatorname{diag}(z_1,\dots,z_n)), \end{align*} then the normalized Weyl integration formula gives \begin{align*} \int_{U(n)}f(g)\,dg=\frac{1}{n!}\int_{(S^1)^n}F(z_1,\dots,z_n)\prod_{1\le i<j\le n}|z_i-z_j|^2\,dt. \end{align*} The factor $\prod_{i<j}|z_i-z_j|^2$ vanishes exactly when two eigenvalues coincide, so the induced eigenvalue density suppresses collisions; this is the circular unitary ensemble density. [/example] The final expression is often the most useful computational form of the theorem. It converts nonabelian integration into abelian integration plus two corrections: quotient by $W$ and multiplication by the square of the Weyl denominator. ## Integration over the Weyl Chamber A final organisational question is whether the factor $1/|W|$ should be viewed as a separate coefficient or as the passage from $T$ to a fundamental domain for $W$. Both views are useful, and the second is closer to the geometric picture of conjugacy classes. [definition: Weyl Alcove in the Torus] Let $\mathfrak t_{\mathrm{ss}}$ be the real span of the coroot directions in $\mathfrak t$, and let $\Lambda_{\mathrm{ss}}=\Lambda\cap \mathfrak t_{\mathrm{ss}}$ for $\Lambda=\ker(\exp:\mathfrak t\to T)$. On the semisimple part, the affine root hyperplanes cut $\mathfrak t_{\mathrm{ss}}$ into alcoves; a Weyl alcove is the closure of one connected component for the affine action generated by $W$ and $\Lambda_{\mathrm{ss}}$. [/definition] If $G$ has a central torus factor, the root hyperplanes do not cut those central directions into bounded alcoves. In that case, a chamber in $T$ means the product of such an alcove image in the semisimple directions with the central torus factor, up to null boundary choices. The chamber description is cleanest away from its boundary. For the integration theorem, the object required is not a particular polytope in all of $\mathfrak t$, but a measurable fundamental domain for the finite $W$-action on $T$. In semisimple examples this is often represented by a closed alcove; in groups such as $U(n)$ the central circle directions remain as torus factors while the roots only order the eigenvalue ratios. Since the boundary maps to singular elements and has measure zero, we need a final form of Weyl integration that replaces the explicit $1/|W|$ by integration over one fundamental region. [quotetheorem:9747] [citeproof:9747] This version is the form used when parametrising conjugacy classes by ordered eigenvalues or by a closed alcove together with any central torus directions. The hypotheses on $C$ are not just technical. If $C$ misses a set of positive measure, the integral undercounts; if two positive-measure pieces of $C$ lie in the same $W$-orbit, the integral overcounts. If the boundary has positive measure, changing the convention for which chamber wall is included can change the value of the restricted integral for an arbitrary integrable representative. The theorem therefore does not say that any visually chosen chamber works; it requires a genuine measurable fundamental domain modulo null boundary ambiguity. With that understood, the chamber form is also the bridge to representation theory. Matrix coefficients and characters are naturally class functions, so their inner products on $G$ reduce to weighted integrals over $T/W$. This is the analytic input behind character orthogonality, and the same Weyl denominator reappears in the Weyl character formula as the alternating factor that compensates for the passage from the torus to the chamber. The Weyl integration formula makes the torus and its Weyl group the correct domain for class-function calculations. From there, the same alternating denominator that appears in the Jacobian becomes the key ingredient in the Weyl character formula, which gives explicit characters from highest weights. # 9. Weyl Character Formula and Consequences This chapter turns the structural results about maximal tori and Weyl groups into explicit formulas for representations. The guiding problem is to recover an irreducible character on a compact connected Lie group from its highest weight. The Weyl character formula answers this by replacing a representation-theoretic object with an alternating finite sum over the Weyl group, and the [Weyl dimension formula](/theorems/9385) extracts dimensions by evaluating the character at the identity in a limiting form. Throughout, let $G$ be a compact connected Lie group with maximal torus $T$, Lie algebra $\mathfrak t$, weight lattice $X^*(T)$, root system $R\subset \mathfrak t^*$, chosen positive roots $R^+$, Weyl group $W=N_G(T)/T$, and dominant integral weights $\Lambda^+$. Write \begin{align*} \rho := \frac{1}{2}\sum_{\alpha\in R^+}\alpha. \end{align*} For a weight $\mu$, write $e^\mu$ for the corresponding formal exponential, or for the corresponding character of $T$ when $\mu$ is integral. Choose a $W$-invariant inner product $(\cdot,\cdot)$ on $\mathfrak t^*$. For a root $\alpha$, its coroot is denoted by $\alpha^\vee$, and pairings with weights are written \begin{align*} \langle \mu,\alpha^\vee\rangle := \frac{2(\mu,\alpha)}{(\alpha,\alpha)}. \end{align*} ## Alternating Sums and the Weyl Denominator The first question is how to build expressions on the torus that remember the sign changes imposed by reflections. Ordinary symmetric sums over $W$ are too coarse for highest weight theory, because they lose the orientation data carried by the choice of positive roots. Alternating sums keep precisely this sign information and are the algebraic source of the denominator in Weyl's formula. [definition: Weyl Alternating Sum] Let $\mu\in \mathfrak t^*$. The Weyl alternating sum attached to $\mu$ is the element \begin{align*} A_\mu := \sum_{w\in W}\det(w)e^{w\mu} \end{align*} of the formal exponential algebra generated by the Weyl orbit of $\mu$. [/definition] When $\mu$ is integral, this finite formal sum may be evaluated as a complex-valued function on $T$. The determinant here is the determinant of $w$ acting on $\mathfrak t^*$. Thus $A_\mu$ changes sign under the Weyl group action rather than remaining invariant. [example: Alternating Sum for SU(2)] For $G=SU(2)$, the Weyl group is $W=\{1,s\}$, where $s\mu=-\mu$, $\det(1)=1$, and $\det(s)=-1$. If $z\in T\cong S^1$ and $e^\mu(z)=z^m$, then $e^{s\mu}(z)=e^{-\mu}(z)=z^{-m}$, so the definition of the Weyl alternating sum gives \begin{align*} A_\mu(z)=\det(1)e^\mu(z)+\det(s)e^{s\mu}(z)=z^m-z^{-m}. \end{align*} Applying the non-trivial reflection sends $z$ to $z^{-1}$, and hence \begin{align*} A_\mu(z^{-1})=(z^{-1})^m-(z^{-1})^{-m}=z^{-m}-z^m=-A_\mu(z). \end{align*} On the reflection fixed set, where $z=z^{-1}$, the same formula gives $A_\mu(z)=z^m-z^m=0$. Thus in rank one the alternating sum changes sign under reflection and vanishes on the reflection wall, just as a sine function does. [/example] The rank-one calculation shows the basic sign behaviour, but the character formula will require a specific alternating expression that can be divided out uniformly in all ranks. The special weight $\rho$ is designed so that the alternating sum $A_\rho$ factors into one contribution from each positive root. The resulting product identity is the [Weyl denominator formula](/theorems/9382), and it supplies the common vanishing factor for all later character numerators. [quotetheorem:9382] [citeproof:9382] This denominator vanishes along the walls of the Weyl chambers, which is exactly what an alternating object should do. The hypotheses matter. In type $A_1$, choosing the positive root $\alpha$ gives $\rho=\alpha/2$ and $A_\rho=e^{\alpha/2}-e^{-\alpha/2}$, while choosing $-\alpha$ instead gives the negative of this expression; the sign convention is harmless only after the same positive system is used everywhere in the quotient. If $G=T$ is itself a torus, then $R^+=\varnothing$, $\rho=0$, and the denominator is $1$, so there is no wall-vanishing phenomenon. The formula is also not a character formula by itself; it identifies the universal alternating factor but says nothing yet about which numerators arise from representations. Its role is forward-looking: once a numerator is known to have the same wall-vanishing behaviour, division by $A_\rho$ can produce a $W$-invariant torus character. [remark: Denominator as a Torus Function] On the torus, the denominator can be written as \begin{align*} \delta(t)=\prod_{\alpha\in R^+}(e^{\alpha/2}(t)-e^{-\alpha/2}(t)). \end{align*} Although individual half-root terms may require passing to a formal cover, the product and the quotient expressions used in the character formula are well-defined as characters on $T$ for compact connected groups in the usual highest-weight setting. [/remark] As Chapter 8 already suggested analytically, the denominator formula also explains why the Weyl integration formula contains the square modulus of a root product. The antisymmetric factor measures the failure of conjugacy classes to intersect the torus transversely at singular points. ## The Weyl Character Formula The main representation-theoretic problem is now the following: given the highest weight $\lambda\in\Lambda^+$ of an irreducible representation $V_\lambda$, can we compute the full character $\chi_\lambda$ without first decomposing all weight spaces? The answer is that the character is the quotient of two alternating Weyl sums. The shift by $\rho$ is the correction that makes the numerator vanish on the same walls as the denominator. [quotetheorem:9384] [citeproof:9384] The formula gives a character from the single input $\lambda$, but its hypotheses are restrictive. Dominance and integrality are necessary: for $SU(2)$, the expression with $\lambda=-\omega$ gives a zero numerator, and a non-integral multiple of $\omega$ does not define a genuine torus character, so neither can be the character of an irreducible representation. Irreducibility is also essential, because a direct sum such as $V_\lambda\oplus V_\mu$ has character $\chi_\lambda+\chi_\mu$ rather than one Weyl quotient attached to a single highest weight. The theorem computes the character once the irreducible highest-weight representation exists; it does not by itself construct the representation or decompose arbitrary tensor products. We still need to justify that this computed class function is enough information to identify the representation. This is where the earlier conjugacy theorem for maximal tori and Schur orthogonality enter: a class function is determined on $T$, and irreducible characters determine irreducible compact-group representations. [quotetheorem:9748] [citeproof:9748] This result is the bridge between highest-weight classification and concrete computation, but it should not be confused with an existence theorem. It says that an irreducible representation, once known to have highest weight $\lambda$, is determined by that weight; construction of such a representation is a separate part of highest-weight theory. The irreducibility hypothesis cannot be dropped: for $SU(2)$, the reducible representations $\operatorname{Sym}^2(\mathbb C^2)\oplus \mathbb C$ and $\operatorname{Sym}^2(\mathbb C^2)\oplus \mathbb C\oplus \mathbb C$ have the same highest weight $2\omega$ in the naive sense of maximal weight, but they are not isomorphic and have different characters. Even knowing only a few weights is not enough in general, since multiplicities and lower-weight patterns distinguish representations that share some visible weight data. Instead of constructing the representation, we may compute its character from $\lambda$ and then read off dimensions, weights, and many tensor-product constraints. [example: SU(2) Character Formula] For $G=SU(2)$, the dominant weights are $m\omega$ with $m\in\mathbb Z_{\ge 0}$, and the single positive root is $2\omega$, so $\rho=\omega$. Put $z=e^\omega(t)$. Since $e^{k\omega}(t)=z^k$ for every integer $k$, the two Weyl alternants are \begin{align*} A_{m\omega+\rho}(t)=A_{(m+1)\omega}(t)=z^{m+1}-z^{-(m+1)}. \end{align*} Also, \begin{align*} A_\rho(t)=A_\omega(t)=z-z^{-1}. \end{align*} The Weyl character formula therefore gives, away from the zeros of $z-z^{-1}$, \begin{align*} \chi_m(t)=\frac{z^{m+1}-z^{-(m+1)}}{z-z^{-1}}. \end{align*} To see the resulting Laurent polynomial, set \begin{align*} S=z^m+z^{m-2}+\cdots+z^{-m}=\sum_{k=0}^{m}z^{m-2k}. \end{align*} Then \begin{align*} zS=\sum_{k=0}^{m}z^{m+1-2k}. \end{align*} Similarly, \begin{align*} z^{-1}S=\sum_{k=0}^{m}z^{m-1-2k}=\sum_{k=1}^{m+1}z^{m+1-2k}. \end{align*} Subtracting cancels the terms with indices $1\le k\le m$, leaving \begin{align*} (z-z^{-1})S=z^{m+1}-z^{-(m+1)}. \end{align*} Hence \begin{align*} \chi_m(t)=z^m+z^{m-2}+\cdots+z^{-m}. \end{align*} This is the character of $\operatorname{Sym}^m(\mathbb C^2)$: on the monomial basis $x^{m-k}y^k$ for $0\le k\le m$, the torus weights are $m-2k$, namely $m,m-2,\dots,-m$, each occurring once. [/example] The $SU(2)$ formula shows why the quotient of alternating expressions becomes an honest character: the denominator divides the numerator in the representation ring. Higher-rank examples behave similarly, but the Weyl group sum contains several reflections and the expansion encodes non-trivial weight multiplicities. [example: SU(3) Standard and Adjoint Characters] Let $G=SU(3)$, and let $t=\operatorname{diag}(z_1,z_2,z_3)\in T$ with $z_1z_2z_3=1$. On the standard basis $e_1,e_2,e_3$ of $\mathbb C^3$, the action is \begin{align*} t e_i=z_i e_i. \end{align*} Therefore the trace of $t$ on the standard representation is \begin{align*} \chi_{\omega_1}(t)=z_1+z_2+z_3. \end{align*} For the dual representation, the action is $(t\varphi)(v)=\varphi(t^{-1}v)$. If $e_i^*$ is the dual basis vector, then \begin{align*} (t e_i^*)(e_i)=e_i^*(t^{-1}e_i)=e_i^*(z_i^{-1}e_i)=z_i^{-1}. \end{align*} Also $(t e_i^*)(e_j)=0$ for $j\ne i$, so $t e_i^*=z_i^{-1}e_i^*$. Hence \begin{align*} \chi_{\omega_2}(t)=z_1^{-1}+z_2^{-1}+z_3^{-1}. \end{align*} For the adjoint representation, $SU(3)$ acts on $\mathfrak{sl}_3(\mathbb C)$ by conjugation. Let $E_{ij}$ be the matrix with a $1$ in the $(i,j)$ entry and zeros elsewhere. For $i\ne j$, \begin{align*} tE_{ij}t^{-1}=z_i z_j^{-1}E_{ij}, \end{align*} because left multiplication by $t$ multiplies row $i$ by $z_i$, while right multiplication by $t^{-1}$ multiplies column $j$ by $z_j^{-1}$. Thus the six off-diagonal root spaces contribute \begin{align*} \sum_{i\ne j} z_i z_j^{-1}. \end{align*} The diagonal traceless subalgebra has basis, for example, $E_{11}-E_{22}$ and $E_{22}-E_{33}$, and each diagonal matrix commutes with $t$, so this two-dimensional subspace contributes $1+1=2$. Therefore the adjoint character, whose highest weight is $\omega_1+\omega_2$, is \begin{align*} \chi_{\omega_1+\omega_2}(t)=2+\sum_{i\ne j}z_i z_j^{-1}. \end{align*} The standard and dual characters record the three coordinate weights, while the adjoint character records the six roots together with the zero weight of multiplicity $2$, the rank of $SU(3)$. [/example] This example separates two sources of weights: roots contribute non-zero weights in the adjoint representation, while the toral part contributes zero weight with multiplicity equal to the rank. The Weyl character formula packages both contributions into a single quotient. ## The Weyl Dimension Formula Once the character is known, the next numerical invariant is the dimension. Formally, $\dim V_\lambda=\chi_\lambda(e)$, where $e\in G$ is the identity. Direct substitution into the Weyl character formula gives $0/0$, so the problem is to evaluate the quotient by taking the leading term near the identity of the torus. [quotetheorem:9385] [citeproof:9385] The formula converts representation theory into finite root-system arithmetic, but the hypotheses again carry real content. The weight must be dominant integral: for $SU(2)$, inserting $\lambda=\frac{1}{2}\omega$ into the product would give $\frac{3}{2}$, not the dimension of any representation, while inserting a non-dominant weight can produce zero or a signed value rather than a dimension. The formula also gives only the total dimension; it does not construct $V_\lambda$, determine the multiplicity of each individual weight, or describe tensor-product decompositions. Its integrality for every dominant integral $\lambda$ is therefore a representation-theoretic consequence, not something visible from the product in isolation, and it points forward to finer multiplicity formulas. [example: Dimension of Symmetric Powers for SU(2)] For $SU(2)$, the positive root is $\alpha=2\omega$ and $\rho=\omega$. Let $\lambda=m\omega$ with $m\in\mathbb Z_{\ge 0}$. By the *Weyl Dimension Formula*, \begin{align*} \dim V_{m\omega}=\frac{(m\omega+\omega,2\omega)}{(\omega,2\omega)}. \end{align*} Since $m\omega+\omega=(m+1)\omega$, bilinearity of the inner product gives \begin{align*} (m\omega+\omega,2\omega)=((m+1)\omega,2\omega)=2(m+1)(\omega,\omega). \end{align*} The denominator is \begin{align*} (\omega,2\omega)=2(\omega,\omega). \end{align*} Because $\omega\ne 0$ and the inner product is positive definite, $(\omega,\omega)\ne 0$, so cancellation gives \begin{align*} \dim V_{m\omega}=\frac{2(m+1)(\omega,\omega)}{2(\omega,\omega)}=m+1. \end{align*} This matches $V_{m\omega}\cong \operatorname{Sym}^m(\mathbb C^2)$: the monomials $x^{m-k}y^k$ for $0\le k\le m$ are indexed by the $m+1$ integers $0,1,\dots,m$, so they form a basis of size $m+1$. [/example] The rank-one calculation is the model for the general case: each positive root contributes one linear factor. In rank two, the interaction between two simple roots and their sum already produces a genuinely two-variable dimension polynomial. [example: $A_2$ Dimension Formula] For type $A_2$, write the dominant highest weight as $\lambda=a\omega_1+b\omega_2$ with $a,b\in\mathbb Z_{\ge 0}$. The positive roots are $\alpha_1$, $\alpha_2$, and $\alpha_1+\alpha_2$, and the half-sum of positive roots is $\rho=\omega_1+\omega_2$. By the defining property of the fundamental weights, \begin{align*} \langle \omega_i,\alpha_j^\vee\rangle=\delta_{ij}. \end{align*} Thus \begin{align*} \lambda+\rho=(a+1)\omega_1+(b+1)\omega_2. \end{align*} Pairing with the first simple coroot gives \begin{align*} \langle \lambda+\rho,\alpha_1^\vee\rangle=(a+1)\langle\omega_1,\alpha_1^\vee\rangle+(b+1)\langle\omega_2,\alpha_1^\vee\rangle=(a+1)\cdot 1+(b+1)\cdot 0=a+1. \end{align*} Similarly, \begin{align*} \langle \lambda+\rho,\alpha_2^\vee\rangle=(a+1)\langle\omega_1,\alpha_2^\vee\rangle+(b+1)\langle\omega_2,\alpha_2^\vee\rangle=(a+1)\cdot 0+(b+1)\cdot 1=b+1. \end{align*} In type $A_2$ all roots have the same length, so \begin{align*} (\alpha_1+\alpha_2)^\vee=\alpha_1^\vee+\alpha_2^\vee. \end{align*} Therefore \begin{align*} \langle \lambda+\rho,(\alpha_1+\alpha_2)^\vee\rangle=\langle \lambda+\rho,\alpha_1^\vee\rangle+\langle \lambda+\rho,\alpha_2^\vee\rangle=(a+1)+(b+1)=a+b+2. \end{align*} For $\rho=\omega_1+\omega_2$, the same pairings are \begin{align*} \langle \rho,\alpha_1^\vee\rangle=\langle\omega_1,\alpha_1^\vee\rangle+\langle\omega_2,\alpha_1^\vee\rangle=1+0=1. \end{align*} Also, \begin{align*} \langle \rho,\alpha_2^\vee\rangle=\langle\omega_1,\alpha_2^\vee\rangle+\langle\omega_2,\alpha_2^\vee\rangle=0+1=1. \end{align*} Finally, \begin{align*} \langle \rho,(\alpha_1+\alpha_2)^\vee\rangle=\langle \rho,\alpha_1^\vee\rangle+\langle \rho,\alpha_2^\vee\rangle=1+1=2. \end{align*} Applying the *Weyl Dimension Formula* in coroot form gives \begin{align*} \dim V_{a\omega_1+b\omega_2}=\frac{\langle\lambda+\rho,\alpha_1^\vee\rangle}{\langle\rho,\alpha_1^\vee\rangle}\cdot\frac{\langle\lambda+\rho,\alpha_2^\vee\rangle}{\langle\rho,\alpha_2^\vee\rangle}\cdot\frac{\langle\lambda+\rho,(\alpha_1+\alpha_2)^\vee\rangle}{\langle\rho,(\alpha_1+\alpha_2)^\vee\rangle}. \end{align*} Substituting the three numerator values and the three denominator values gives \begin{align*} \dim V_{a\omega_1+b\omega_2}=\frac{a+1}{1}\cdot\frac{b+1}{1}\cdot\frac{a+b+2}{2}=\frac{(a+1)(b+1)(a+b+2)}{2}. \end{align*} For $(a,b)=(1,0)$, this gives \begin{align*} \dim V_{\omega_1}=\frac{(1+1)(0+1)(1+0+2)}{2}=\frac{2\cdot 1\cdot 3}{2}=3. \end{align*} For $(a,b)=(1,1)$, this gives \begin{align*} \dim V_{\omega_1+\omega_2}=\frac{(1+1)(1+1)(1+1+2)}{2}=\frac{2\cdot 2\cdot 4}{2}=8. \end{align*} Thus the formula recovers the dimensions of the standard representation and the adjoint representation of $SU(3)$ from the three positive-root factors of $A_2$. [/example] These computations are the first evidence that the Weyl formulas are not merely existence theorems. They give practical closed forms for characters and dimensions from the combinatorics of the root system. [remark: What the Formula Does Not Directly Give] The Weyl character formula determines the full character, but extracting individual weight multiplicities from the quotient may still require expansion. Later tools, such as Kostant's multiplicity formula or Freudenthal recursion, refine the same highest-weight data into explicit multiplicities. [/remark] The chapter therefore completes the passage from compact Lie group structure to representation-theoretic calculation. Maximal tori reduce characters to finite Fourier sums, Weyl groups impose the alternating symmetry, and the shift by $\rho$ produces the exact correction needed for irreducible highest-weight representations. In type $A$, this is the same mechanism that turns alternants into Schur polynomials, so the Weyl character formula also connects compact-group representation theory with symmetric polynomials and Fourier analysis on tori. The Weyl character formula ties together roots, weights, chambers, and characters in a single explicit expression. The final chapter uses that structure to explain how compact connected Lie groups are assembled from their classical families, showing how the invariants developed earlier determine the group itself. # 10. Compact Structure and Classical Families Chapters 2 through 9 developed the maximal torus, Weyl group, roots, weights, integration formula, and characters of a compact connected Lie group. This final structural chapter explains how these invariants assemble the group itself, not merely its representations. The central message is that compact connected Lie groups are built from a torus and a simply connected compact semisimple group, with only a finite central identification left to record. We then survey the Dynkin classification and close by separating the connected theory from the extra data needed for disconnected compact groups. ## Splitting a Compact Connected Group into Toral and Semisimple Pieces What part of a compact connected Lie group is abelian, and what part carries the root-theoretic structure studied earlier? The Lie algebra answers this first: an invariant inner product splits off the centre, while the derived algebra is compact semisimple. The global group is slightly subtler because products at the Lie algebra level may overlap by a finite central subgroup. [definition: Semisimple Part of a Compact Connected Lie Group] Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$. The semisimple part of $G$ is the connected Lie subgroup $G_{\mathrm{ss}} \le G$ with Lie algebra \begin{align*} [\mathfrak g,\mathfrak g] \subset \mathfrak g. \end{align*} [/definition] This subgroup contains the non-abelian root directions. The remaining directions are central, so they exponentiate to a compact connected abelian Lie group. [definition: Central Torus] Let $G$ be a compact connected Lie group with centre $Z(G)$. The central torus of $G$ is the identity component $Z(G)^\circ$. [/definition] The two pieces interact through a finite overlap. The central torus and the semisimple subgroup are not usually disjoint factors: they may share finite central elements, and quotienting by that overlap is the obstruction to writing $G$ as a direct product. The structural question is whether this finite central ambiguity is the only way the abelian and semisimple parts can combine in a compact connected group. [quotetheorem:9750] [citeproof:9750] The theorem reduces many questions about compact connected groups to two ingredients: a torus, which is controlled by lattices, and a simply connected semisimple group, which is controlled by roots. The finite subgroup $F$ records which central elements of the two factors have been identified. Each hypothesis in the statement is doing work. Compactness is used twice: it gives an $\operatorname{Ad}(G)$-invariant inner product by averaging over $G$, and it forces the discrete kernel of the covering map to be finite. For a noncompact connected group such as $SL(2,\mathbb R)$, the maximal compact torus and the noncompact directions do not assemble into a quotient of a compact torus times a compact simply connected semisimple group. Connectedness is also essential: $O(n)$ is compact but disconnected, and its extra component is not recovered from the central quotient presentation of $SO(n)^\circ$ alone. The word central is the remaining boundary of the theorem. The quotient subgroup $F$ is central because it is the kernel of a covering homomorphism from the product of the toral and semisimple factors. The theorem therefore classifies the connected group up to finite central identifications between those factors; it does not classify all extensions by a finite group, all disconnected compact groups, or noncentral semidirect products. [example: Unitary Groups as Central Quotients] Define \begin{align*} \phi:S^1\times SU(n)\to U(n),\qquad \phi(z,A)=zA. \end{align*} This is a homomorphism because scalar matrices commute with every matrix: \begin{align*} \phi(z,A)\phi(w,B)=(zA)(wB)=zwAB=\phi(zw,AB). \end{align*} To prove surjectivity, let $B\in U(n)$. Since $\det(B)\in S^1$, choose $z\in S^1$ with $z^n=\det(B)$, and set $A=z^{-1}B$. Then $A$ is unitary, and \begin{align*} \det(A)=\det(z^{-1}B)=\det(z^{-1}I)\det(B)=z^{-n}\det(B)=1. \end{align*} Thus $A\in SU(n)$, and \begin{align*} \phi(z,A)=z(z^{-1}B)=B. \end{align*} Now compute the kernel. If $(z,A)\in\ker(\phi)$, then $zA=I$, hence $A=z^{-1}I$. The condition $A\in SU(n)$ gives \begin{align*} 1=\det(A)=\det(z^{-1}I)=z^{-n}, \end{align*} so $z^n=1$. Conversely, if $\zeta^n=1$, then $\zeta^{-1}I\in SU(n)$ and \begin{align*} \phi(\zeta,\zeta^{-1}I)=\zeta(\zeta^{-1}I)=I. \end{align*} Therefore \begin{align*} \ker(\phi)=\{(\zeta,\zeta^{-1}I):\zeta^n=1\}. \end{align*} This kernel is a copy of $\mu_n$, embedded in the centre of $S^1\times SU(n)$ by $\zeta\mapsto(\zeta,\zeta^{-1}I)$. Hence \begin{align*} U(n)\cong (S^1\times SU(n))/\mu_n. \end{align*} The Lie algebra splitting separates the scalar direction from $\mathfrak{su}(n)$, while the group remembers that the two factors share the finite subgroup of scalar $n$th roots of unity. [/example] This example is the model for the general finite central quotient. The Lie algebra sees $\mathfrak u(n)=i\mathbb R\oplus\mathfrak{su}(n)$, while the group remembers the shared scalar matrices. [example: Spin Cover of the Special Orthogonal Group] For $n\ge 3$, let \begin{align*} p:\operatorname{Spin}(n)\to SO(n) \end{align*} be the standard spin covering. Its kernel is \begin{align*} \ker(p)=\{1,-1\}. \end{align*} Since $p$ is surjective, every element of $SO(n)$ is $p(s)$ for some $s\in \operatorname{Spin}(n)$. The quotient map therefore identifies two elements $s,t\in \operatorname{Spin}(n)$ exactly when \begin{align*} p(s)=p(t). \end{align*} This equality is equivalent to \begin{align*} p(t^{-1}s)=p(t)^{-1}p(s)=I, \end{align*} so \begin{align*} t^{-1}s\in \ker(p)=\{1,-1\}. \end{align*} Thus either $s=t$ or $s=-t$, and the fibres of $p$ are precisely the cosets of $\{\pm1\}$. Hence the induced map \begin{align*} \operatorname{Spin}(n)/\{\pm1\}\to SO(n) \end{align*} is a bijective Lie group homomorphism, so \begin{align*} SO(n)\cong \operatorname{Spin}(n)/\{\pm1\}. \end{align*} For $n=2m+1$, the Lie algebra $\mathfrak{so}(2m+1)$ has root system of type $B_m$, while for $n=2m$, the Lie algebra $\mathfrak{so}(2m)$ has root system of type $D_m$. Thus $SO(n)$ is obtained from the simply connected compact group of the corresponding type by quotienting by the central subgroup $\{\pm1\}$. [/example] The distinction between simply connected and adjoint forms is invisible at the level of real Lie algebras but visible in representation theory. It determines which weights occur as highest weights of honest representations of the group. ## Dynkin Diagrams and the Classical Simply Connected Groups Once the compact connected group is reduced to the simply connected semisimple case, what data classifies it? A maximal torus gives a root system, and the choice of Weyl chamber gives simple roots. The incidence pattern among these simple roots is encoded by a Dynkin diagram. [definition: Dynkin Diagram of a Root System] Let $R$ be a reduced root system in a Euclidean vector space $E$, and let $\Delta=\{\alpha_1,\dots,\alpha_r\}$ be a base of simple roots. The Dynkin diagram has one vertex for each $\alpha_i$. For $i\ne j$, the number of edges and the direction of the arrow are determined by the Cartan integers \begin{align*} \langle \alpha_i,\alpha_j^\vee\rangle = \frac{2(\alpha_i,\alpha_j)}{(\alpha_j,\alpha_j)}. \end{align*} [/definition] The diagram packages the angles and relative lengths of simple roots. To turn this combinatorial picture into a classification of groups, we must know that no information has been lost: the diagram reconstructs the root datum, and the root datum determines the simply connected compact semisimple group. The next theorem records that passage from geometry to classification, while separating the fully proved compact structure theorem from the larger classification theorem quoted from semisimple Lie theory. [quotetheorem:9751] This classification is usually proved through the complex semisimple Lie algebra classification and the integration theorem for compact real forms. In this course we use it as a structural endpoint rather than reproving the full existence and uniqueness theorem. Root systems by themselves are not enough: $SU(2)$ and $SO(3)$ have the same type $A_1$ root system, but their weight lattices differ by the quotient $P/Q\cong\mathbb Z/2\mathbb Z$. [example: Type A] For $n\ge 2$, take the maximal torus of $SU(n)$ to be the diagonal unitary matrices of determinant $1$. Its Lie algebra consists of \begin{align*} H=\operatorname{diag}(it_1,\dots,it_n)\quad \text{with}\quad t_1+\cdots+t_n=0. \end{align*} Let $E_{ij}$ be the matrix with $1$ in the $(i,j)$-entry and $0$ elsewhere. For $i\ne j$, the matrices $E_{ij}$ span the off-diagonal root spaces in $\mathfrak{sl}_n(\mathbb C)$. Since $H E_{ij}$ has $(i,j)$-entry $it_i$ and $E_{ij}H$ has $(i,j)$-entry $it_j$, we get \begin{align*} [H,E_{ij}]=H E_{ij}-E_{ij}H=i(t_i-t_j)E_{ij}. \end{align*} Thus the corresponding root is \begin{align*} \alpha_{ij}(H)=t_i-t_j,\qquad i\ne j. \end{align*} Choose the positive chamber $t_1\ge t_2\ge \cdots \ge t_n$. Then the positive roots are $\alpha_{ij}=t_i-t_j$ with $i<j$. For $i<j$, \begin{align*} t_i-t_j=(t_i-t_{i+1})+(t_{i+1}-t_{i+2})+\cdots+(t_{j-1}-t_j). \end{align*} Hence every positive root is a nonnegative integer combination of \begin{align*} \alpha_1=t_1-t_2,\quad \alpha_2=t_2-t_3,\quad \dots,\quad \alpha_{n-1}=t_{n-1}-t_n. \end{align*} None of these $\alpha_i$ can be written as a sum of two positive roots, because $\alpha_i=t_i-t_{i+1}$ has no index strictly between $i$ and $i+1$. Therefore the simple roots are exactly $\alpha_i=t_i-t_{i+1}$ for $1\le i\le n-1$. Identifying $\alpha_i$ with $e_i-e_{i+1}$ in the hyperplane $t_1+\cdots+t_n=0$, the inherited Euclidean inner product gives \begin{align*} (\alpha_i,\alpha_i)=(e_i-e_{i+1},e_i-e_{i+1})=1+1=2. \end{align*} If $|i-j|=1$, the two roots share exactly one basis vector with opposite signs, so $(\alpha_i,\alpha_j)=-1$. If $|i-j|>1$, they involve disjoint basis vectors, so $(\alpha_i,\alpha_j)=0$. Since all simple roots have squared length $2$, the Cartan integer is \begin{align*} \langle \alpha_i,\alpha_j^\vee\rangle=\frac{2(\alpha_i,\alpha_j)}{(\alpha_j,\alpha_j)}=(\alpha_i,\alpha_j). \end{align*} Thus adjacent vertices are joined by one edge, non-adjacent vertices are not joined, and no arrows occur. The Dynkin diagram is therefore a chain of $n-1$ vertices, the diagram of type $A_{n-1}$. [/example] Type $A$ is the first place where the centre matters. The centre of $SU(n)$ is $\mu_n$, and quotienting by subgroups of $\mu_n$ changes the allowed weight lattice. [example: Types B and D] Let $e_1,\dots,e_m$ be the standard orthonormal basis of $\mathbb R^m$, so $(e_i,e_j)=\delta_{ij}$. For type $B_m$, the roots are \begin{align*} B_m=\{\pm e_i,\ \pm e_i\pm e_j:i\ne j\}. \end{align*} The roots $\pm e_i$ have squared length \begin{align*} (e_i,e_i)=1, \end{align*} while the roots $\pm e_i\pm e_j$ with $i\ne j$ have squared length \begin{align*} (e_i\pm e_j,e_i\pm e_j)=(e_i,e_i)\pm 2(e_i,e_j)+(e_j,e_j)=1+0+1=2. \end{align*} Thus $B_m$ has short roots $\pm e_i$ and long roots $\pm e_i\pm e_j$. Choose simple roots \begin{align*} \alpha_1=e_1-e_2,\quad \alpha_2=e_2-e_3,\quad \dots,\quad \alpha_{m-1}=e_{m-1}-e_m,\quad \alpha_m=e_m. \end{align*} For $1\le i\le m-1$, \begin{align*} (\alpha_i,\alpha_i)=(e_i-e_{i+1},e_i-e_{i+1})=1-0-0+1=2, \end{align*} and \begin{align*} (\alpha_m,\alpha_m)=(e_m,e_m)=1. \end{align*} The last adjacent inner product is \begin{align*} (\alpha_{m-1},\alpha_m)=(e_{m-1}-e_m,e_m)=0-1=-1. \end{align*} Therefore \begin{align*} \langle \alpha_{m-1},\alpha_m^\vee\rangle=\frac{2(\alpha_{m-1},\alpha_m)}{(\alpha_m,\alpha_m)}=\frac{2(-1)}{1}=-2, \end{align*} while \begin{align*} \langle \alpha_m,\alpha_{m-1}^\vee\rangle=\frac{2(\alpha_m,\alpha_{m-1})}{(\alpha_{m-1},\alpha_{m-1})}=\frac{2(-1)}{2}=-1. \end{align*} This unequal pair of Cartan integers gives the double edge at the end of the $B_m$ Dynkin diagram, with the arrow pointing toward the shorter root $\alpha_m$. For type $D_m$, the roots are \begin{align*} D_m=\{\pm e_i\pm e_j:i\ne j\}. \end{align*} Every root has squared length $2$, since for $i\ne j$, \begin{align*} (e_i\pm e_j,e_i\pm e_j)=1+0+1=2. \end{align*} A convenient system of simple roots is \begin{align*} \alpha_1=e_1-e_2,\quad \dots,\quad \alpha_{m-1}=e_{m-1}-e_m,\quad \alpha_m=e_{m-1}+e_m. \end{align*} The two final roots both meet $\alpha_{m-2}=e_{m-2}-e_{m-1}$: \begin{align*} (\alpha_{m-2},\alpha_{m-1})=(e_{m-2}-e_{m-1},e_{m-1}-e_m)=-1, \end{align*} and \begin{align*} (\alpha_{m-2},\alpha_m)=(e_{m-2}-e_{m-1},e_{m-1}+e_m)=-1. \end{align*} But the two final roots are orthogonal to each other: \begin{align*} (\alpha_{m-1},\alpha_m)=(e_{m-1}-e_m,e_{m-1}+e_m)=1-1=0. \end{align*} Since all simple roots in $D_m$ have squared length $2$, each nonzero adjacent inner product $-1$ gives a single edge. Thus the diagram is a chain until $\alpha_{m-2}$, where it branches to the two end vertices $\alpha_{m-1}$ and $\alpha_m$. The simply connected compact groups with these root systems are $\operatorname{Spin}(2m+1)$ for type $B_m$ and $\operatorname{Spin}(2m)$ for type $D_m$. The covering homomorphisms \begin{align*} \operatorname{Spin}(2m+1)\to SO(2m+1) \end{align*} and \begin{align*} \operatorname{Spin}(2m)\to SO(2m) \end{align*} have kernel $\{1,-1\}$. Passing to $SO(2m+1)$ or $SO(2m)$ therefore identifies $s$ with $-s$ in the corresponding spin group. Representations on which $-1$ acts by $-1$, including the spin representations, cannot factor through this quotient; this is the representation-theoretic effect of the central identification. [/example] The root systems show the geometric difference between odd and even orthogonal groups. In type $B_m$ short roots appear; in type $D_m$ all roots have the same length and the diagram branches at the end. [example: Type C] The simply connected compact group of type $C_m$ is the compact symplectic group $Sp(m)$. In the standard Euclidean space $\mathbb R^m$ with orthonormal basis $e_1,\dots,e_m$, its root system may be realised as \begin{align*} C_m=\{\pm 2e_i,\ \pm e_i\pm e_j:i\ne j\}. \end{align*} The roots of the form $\pm 2e_i$ have squared length \begin{align*} (2e_i,2e_i)=4(e_i,e_i)=4. \end{align*} For $i\ne j$, the roots of the form $\pm e_i\pm e_j$ have squared length \begin{align*} (e_i\pm e_j,e_i\pm e_j)=(e_i,e_i)\pm 2(e_i,e_j)+(e_j,e_j)=1+0+1=2. \end{align*} Thus $\pm 2e_i$ are the long roots, while $\pm e_i\pm e_j$ are the short roots. A standard choice of simple roots is \begin{align*} \alpha_1=e_1-e_2,\quad \alpha_2=e_2-e_3,\quad \dots,\quad \alpha_{m-1}=e_{m-1}-e_m,\quad \alpha_m=2e_m. \end{align*} For $1\le i\le m-1$, \begin{align*} (\alpha_i,\alpha_i)=(e_i-e_{i+1},e_i-e_{i+1})=1-0-0+1=2, \end{align*} while \begin{align*} (\alpha_m,\alpha_m)=(2e_m,2e_m)=4. \end{align*} The last adjacent inner product is \begin{align*} (\alpha_{m-1},\alpha_m)=(e_{m-1}-e_m,2e_m)=0-2=-2. \end{align*} Therefore \begin{align*} \langle \alpha_{m-1},\alpha_m^\vee\rangle=\frac{2(\alpha_{m-1},\alpha_m)}{(\alpha_m,\alpha_m)}=\frac{2(-2)}{4}=-1, \end{align*} whereas \begin{align*} \langle \alpha_m,\alpha_{m-1}^\vee\rangle=\frac{2(\alpha_m,\alpha_{m-1})}{(\alpha_{m-1},\alpha_{m-1})}=\frac{2(-2)}{2}=-2. \end{align*} So the $C_m$ Dynkin diagram has a double edge at the end, with the arrow pointing toward the shorter root $\alpha_{m-1}$. This is opposite to type $B_m$: in type $B_m$ the final simple root is short, while in type $C_m$ the final simple root $2e_m$ is long. The Weyl group is nevertheless the same signed permutation group, because permuting the basis vectors and changing signs sends the set $\{\pm 2e_i,\ \pm e_i\pm e_j:i\ne j\}$ to itself. [/example] These classical families account for the matrix groups most often encountered in geometry. The exceptional diagrams $G_2,F_4,E_6,E_7,E_8$ complete the semisimple classification, but the classical families already display the main features: centres, covers, root lengths, Weyl groups, and highest-weight restrictions. [remark: Lie Algebra Type Does Not Determine the Group] The compact groups $SU(2)$ and $SO(3)$ have isomorphic Lie algebras, but they are not isomorphic as Lie groups. The covering $SU(2)\to SO(3)$ has kernel $\{\pm I\}$, and representations of $SO(3)$ are exactly the representations of $SU(2)$ on which $-I$ acts as the identity. [/remark] This is why the classification must remember lattices, not only roots. Roots describe the adjoint action, while weights describe possible characters of the maximal torus in representations. ## Lattices, Centres, and Allowed Highest Weights How does the finite central quotient change representation theory? The highest-weight theorem says that irreducible representations of a simply connected compact semisimple group are indexed by dominant integral weights. Passing to a quotient removes exactly those representations on which the kernel acts by a non-identity scalar. [definition: Root Lattice and Weight Lattice] Let $G$ be a compact connected semisimple Lie group with maximal torus $T$ and root system $R$. The root lattice is \begin{align*} Q = \operatorname{span}_{\mathbb Z}(R). \end{align*} For the simply connected form, the weight lattice is \begin{align*} P = \{\lambda : \lambda(\alpha^\vee)\in\mathbb Z \text{ for every root }\alpha\in R\}. \end{align*} [/definition] The quotient $P/Q$ measures how far the simply connected group is from its adjoint form. It is not just a bookkeeping device: its character group is the centre, and central characters decide whether a highest-weight representation survives a quotient. The next theorem makes this relationship explicit and gives the descent criterion used in examples such as $SU(2)\to SO(3)$. [quotetheorem:9752] [citeproof:9752] The theorem gives a practical test: compute the highest weight modulo the root lattice and check whether the chosen central subgroup is killed. The simply connected hypothesis is what makes the character lattice of the maximal torus equal to the full weight lattice $P$. If the group is already a quotient of $K$, its maximal torus has a smaller character lattice between $Q$ and $P$, so some dominant integral weights for the Lie algebra no longer define representations of that group. For a non-simply connected compact semisimple group $G$, the same root system still describes the infinitesimal adjoint action, but the representation theory is obtained by lifting to the simply connected cover $K$ and imposing a descent condition on the kernel. Thus the criterion above tests central quotients of the simply connected form. It does not decide descent for arbitrary homomorphisms from unrelated groups, and it does not classify disconnected extensions where a component group may also permute highest weights by outer automorphisms. [example: Comparing SU(2) and SO(3)] For $SU(2)$, let $\omega$ be the fundamental weight. The unique positive root is \begin{align*} \alpha=2\omega. \end{align*} Hence the weight lattice and root lattice are \begin{align*} P=\mathbb Z\omega,\qquad Q=\mathbb Z\alpha=\mathbb Z(2\omega)=2\mathbb Z\omega. \end{align*} Therefore \begin{align*} P/Q=\mathbb Z\omega/2\mathbb Z\omega\cong \mathbb Z/2\mathbb Z. \end{align*} The dominant weights are $m\omega$ with $m\in\mathbb Z_{\ge 0}$. In the irreducible representation of highest weight $m\omega$, the rank-one highest-weight string is \begin{align*} m\omega,\ (m\omega-\alpha),\ (m\omega-2\alpha),\ \dots,\ (m\omega-m\alpha). \end{align*} Since $\alpha=2\omega$, these weights are \begin{align*} m\omega,\ (m-2)\omega,\ (m-4)\omega,\ \dots,\ -m\omega. \end{align*} There are $m+1$ terms, and each weight space in this $SU(2)$ string is one-dimensional, so the representation has dimension $m+1$. Now write the maximal torus element of $SU(2)$ as \begin{align*} t(\theta)=\operatorname{diag}(e^{i\theta},e^{-i\theta}). \end{align*} The highest-weight line of weight $m\omega$ is acted on by the character \begin{align*} t(\theta)\mapsto e^{im\theta}. \end{align*} The central element $-I$ is $t(\pi)$, because \begin{align*} t(\pi)=\operatorname{diag}(e^{i\pi},e^{-i\pi})=\operatorname{diag}(-1,-1)=-I. \end{align*} Thus $-I$ acts on the highest-weight line by \begin{align*} e^{im\pi}=(-1)^m. \end{align*} Since $-I$ is central, it acts by the same scalar on the whole irreducible representation. A representation of $SU(2)$ factors through the quotient $SU(2)/\{\pm I\}$ exactly when every element of $\{\pm I\}$ acts as the identity. Therefore the irreducible representations descending to \begin{align*} SO(3)=SU(2)/\{\pm I\} \end{align*} are precisely those with $(-1)^m=1$, namely those with even $m$. [/example] This example explains the familiar split between integer and half-integer spin. The Lie algebra allows all finite-dimensional $\mathfrak{sl}_2(\mathbb C)$-modules, while the group $SO(3)$ only sees the even highest weights. [example: Fundamental Weights for SU(3)] For $SU(3)$, work in the plane $t_1+t_2+t_3=0$ and identify the simple roots with \begin{align*} \alpha_1=e_1-e_2,\qquad \alpha_2=e_2-e_3. \end{align*} Their squared lengths are \begin{align*} (\alpha_1,\alpha_1)=(e_1-e_2,e_1-e_2)=1+1=2 \end{align*} and \begin{align*} (\alpha_2,\alpha_2)=(e_2-e_3,e_2-e_3)=1+1=2. \end{align*} Hence $\alpha_i^\vee=\alpha_i$ for $i=1,2$. The fundamental weights $\omega_1,\omega_2$ are defined by $\omega_i(\alpha_j^\vee)=\delta_{ij}$. In this normalization, they are \begin{align*} \omega_1=\frac{2e_1-e_2-e_3}{3},\qquad \omega_2=\frac{e_1+e_2-2e_3}{3}. \end{align*} Indeed, \begin{align*} (\omega_1,\alpha_1)=\frac{1}{3}(2e_1-e_2-e_3,e_1-e_2)=\frac{1}{3}(2+1)=1, \end{align*} while \begin{align*} (\omega_1,\alpha_2)=\frac{1}{3}(2e_1-e_2-e_3,e_2-e_3)=\frac{1}{3}(-1+1)=0. \end{align*} Similarly, \begin{align*} (\omega_2,\alpha_1)=\frac{1}{3}(e_1+e_2-2e_3,e_1-e_2)=\frac{1}{3}(1-1)=0 \end{align*} and \begin{align*} (\omega_2,\alpha_2)=\frac{1}{3}(e_1+e_2-2e_3,e_2-e_3)=\frac{1}{3}(1+2)=1. \end{align*} Every dominant weight has the form \begin{align*} \lambda=a\omega_1+b\omega_2 \end{align*} with $a,b\in\mathbb Z_{\ge0}$. The simple roots expressed in the fundamental-weight basis are \begin{align*} \alpha_1=2\omega_1-\omega_2 \end{align*} and \begin{align*} \alpha_2=-\omega_1+2\omega_2. \end{align*} Thus the root lattice is \begin{align*} Q=\mathbb Z(2\omega_1-\omega_2)+\mathbb Z(-\omega_1+2\omega_2) \end{align*} inside the weight lattice \begin{align*} P=\mathbb Z\omega_1+\mathbb Z\omega_2. \end{align*} Modulo $Q$, the relation $\alpha_1=0$ gives $\omega_2=2\omega_1$, and the relation $\alpha_2=0$ then gives \begin{align*} 0=-\omega_1+2\omega_2=-\omega_1+4\omega_1=3\omega_1. \end{align*} Therefore \begin{align*} P/Q\cong \mathbb Z/3\mathbb Z, \end{align*} generated by the class of $\omega_1$. For $\lambda=a\omega_1+b\omega_2$, using $\omega_2=2\omega_1$ modulo $Q$ gives \begin{align*} \lambda=a\omega_1+b\omega_2\equiv a\omega_1+2b\omega_1=(a+2b)\omega_1 \pmod Q. \end{align*} The centre of $SU(3)$ is \begin{align*} Z(SU(3))=\{\zeta I:\zeta^3=1\}\cong \mu_3. \end{align*} Thus the residue class of $a+2b$ modulo $3$ determines the central character: the representation of highest weight $a\omega_1+b\omega_2$ has identity central character exactly when \begin{align*} a+2b\equiv 0 \pmod 3. \end{align*} Geometrically, the dominant chamber is the lattice cone spanned by $\omega_1$ and $\omega_2$, and passing from $SU(3)$ to a central quotient keeps only those lattice points whose class modulo the root lattice is killed by the quotient. [/example] The diagram, the chamber, and the lattice picture are three views of the same classification data. For computations with characters, the dominant lattice points are the most concrete version. ## Disconnected Compact Groups and Diagram Automorphisms What changes when the compact Lie group is no longer connected? The identity component still has the structure described above, but the remaining components act on it by conjugation. This produces finite symmetry data, often visible as automorphisms of the Dynkin diagram. [definition: Component Group] Let $G$ be a compact Lie group. The component group of $G$ is \begin{align*} \pi_0(G)=G/G^\circ, \end{align*} where $G^\circ$ is the identity component. [/definition] Since $G$ is compact, the component group is finite when $G$ is a compact Lie group. Its action by conjugation may be outer on $G^\circ$, so we need a way to record the resulting symmetry of the connected classification data. [definition: Diagram Automorphism] Let $G^\circ$ be compact connected semisimple with chosen maximal torus and simple roots, and let $\Gamma$ be the associated Dynkin diagram. A diagram automorphism is a graph isomorphism \begin{align*} \sigma:\Gamma\longrightarrow\Gamma \end{align*} from the Dynkin diagram to itself preserving the Cartan matrix data. [/definition] Diagram automorphisms describe the outer symmetries of the root datum. The next theorem explains why these finite symmetries enter disconnected compact groups: each component conjugates the identity component, and after quotienting by inner automorphisms this action lands in the automorphism group of the root datum. [quotetheorem:9754] [citeproof:9754] The theorem is a warning about classification: for disconnected groups, the connected classification is only the first layer. One must also specify how the finite component group acts. The simplest boundary case is the comparison between $SO(n)$ and $O(n)$: they share the same identity component when viewed inside $O(n)$, but $O(n)$ has an additional component represented by a reflection, and conjugation by that reflection can act in a non-identity way on the connected data. The theorem also does not claim that the outer action alone reconstructs the disconnected group. Extension data can remain after the homomorphism $\pi_0(G)\to\operatorname{Out}(G^\circ)$ is fixed: different lifts of the same outer action may differ by central cocycles or by a non-split extension of the component group by $G^\circ$. The statement identifies the first invariant that must be added to the connected classification, not a complete classification of all disconnected compact Lie groups. [example: Orthogonal Groups] For $n\ge 2$, the determinant map separates the two components of $O(n)$. Since every orthogonal matrix has determinant $\pm 1$, there is a homomorphism \begin{align*} \det:O(n)\to \{\pm 1\}. \end{align*} Its kernel is \begin{align*} \ker(\det)=\{A\in O(n):\det(A)=1\}=SO(n). \end{align*} The reflection \begin{align*} r=\operatorname{diag}(-1,1,\dots,1) \end{align*} has determinant $-1$, so every element of $O(n)$ lies either in $SO(n)$ or in $rSO(n)$. Thus \begin{align*} O(n)/SO(n)\cong \{\pm1\}\cong \mathbb Z/2\mathbb Z. \end{align*} Conjugation by $r$ preserves $SO(n)$ because, for $A\in SO(n)$, \begin{align*} \det(rAr^{-1})=\det(r)\det(A)\det(r^{-1})=(-1)(1)(-1)=1. \end{align*} Hence $A\mapsto rAr^{-1}$ is an automorphism of $SO(n)$ coming from the non-identity component of $O(n)$. Now take $n=2m$ and use the standard maximal torus in $SO(2m)$ consisting of rotations in the coordinate planes. Write \begin{align*} R(\theta)=\begin{pmatrix} \cos\theta & -\sin\theta ; \sin\theta & \cos\theta \end{pmatrix}, \end{align*} where the semicolon separates the two rows. A torus element has rotation blocks $R(\theta_1),\dots,R(\theta_m)$. In the last coordinate plane, conjugating by the reflection $s=\operatorname{diag}(1,-1)$ gives \begin{align*} sR(\theta_m)s^{-1}=R(-\theta_m). \end{align*} Indeed, multiplying on the left changes the sign of the second row, and multiplying on the right changes the sign of the second column, giving entries $\cos\theta_m$, $\sin\theta_m$, $-\sin\theta_m$, and $\cos\theta_m$, which are exactly the entries of $R(-\theta_m)$. Thus the induced action on the usual coordinates of the type $D_m$ root system fixes $e_1,\dots,e_{m-1}$ and sends $e_m$ to $-e_m$. For the standard simple roots \begin{align*} \alpha_1=e_1-e_2,\quad \dots,\quad \alpha_{m-1}=e_{m-1}-e_m,\quad \alpha_m=e_{m-1}+e_m, \end{align*} this action fixes $\alpha_1,\dots,\alpha_{m-2}$. It sends \begin{align*} \alpha_{m-1}=e_{m-1}-e_m \end{align*} to \begin{align*} e_{m-1}+e_m=\alpha_m, \end{align*} and sends \begin{align*} \alpha_m=e_{m-1}+e_m \end{align*} to \begin{align*} e_{m-1}-e_m=\alpha_{m-1}. \end{align*} So for type $D_m$ with $m\ge 5$, the non-identity component of $O(2m)$ acts by the nontrivial Dynkin diagram symmetry exchanging the two terminal vertices. In the low-rank cases, the same root computation remains valid, but the diagram has extra coincidences: $D_3\cong A_3$, and $D_4$ has the larger triality symmetry group. [/example] This example shows why disconnected compact groups naturally lead beyond root systems to equivariant root data. The extra component may identify representations or twist them by an outer automorphism. [remark: What the Classification Has Achieved] For compact connected groups, the main ambiguity after the Lie algebra is finite and central. For disconnected compact groups, a finite group of components may act in a way not generated by conjugation inside the identity component. Thus the compact theory separates into continuous data, captured by tori and semisimple root systems, and finite data, captured by central quotients and component actions. [/remark] The course can now use this structure theorem as a dictionary. Matrix groups such as $U(n)$, $SU(n)$, $SO(n)$, $\operatorname{Spin}(n)$, and $Sp(n)$ become examples of a common pattern, while representation-theoretic tools such as Weyl integration and the Weyl character formula read the same data from the torus, Weyl group, and dominant weights. ## Beyond and Connections This note is meant to sit between several public Androma themes. The topological and measure-theoretic parts connect to [Lie Groups I: Foundations](/page/Lie%20Groups%20I%3A%20Foundations), [Topological Group Structure](/theorems/2505), [Compact Space](/page/Compact%20Space), [Hilbert Space](/page/Hilbert%20Space), and [Existence and Uniqueness of Haar Measure](/theorems/1063). The infinitesimal chapters connect to [Lie Algebras I: Foundations](/page/Lie%20Algebras%20I%3A%20Foundations), [Lie Algebras II: Structure and Classification](/page/Lie%20Algebras%20II%3A%20Structure%20and%20Classification), [Cartan Subalgebras of Semisimple Lie Algebras over Algebraically Closed Fields Are Abelian](/theorems/4682), [Chambers and Positive Systems of a Root System](/theorems/4698), and [Coxeter Presentation of the Weyl Group](/theorems/4707). The final representation-theoretic chapters connect to [Cambridge II Representation Theory](/page/Cambridge%20II%20Representation%20Theory), [Complete Reducibility Theorem for Compact Lie Groups](/theorems/8826), [Characters Determine Representations](/theorems/2425), [Classification of Finite-Dimensional Irreducible Highest Weight Modules](/theorems/9373), and [Peter--Weyl Theorem](/theorems/8833). Several directions naturally continue from here. One can specialize the classification theorem to the classical matrix groups using [Lie Groups I: Foundations](/page/Lie%20Groups%20I%3A%20Foundations), use the Weyl Integration Formula together with [Existence and Uniqueness of Haar Measure](/theorems/1063) to compute explicit character and volume formulas, compare compact groups with their complex reductive [Lie Algebras I: Foundations](/page/Lie%20Algebras%20I%3A%20Foundations), or study how dominant weights classify irreducible representations through [Cambridge II Representation Theory](/page/Cambridge%20II%20Representation%20Theory). For disconnected compact Lie groups, the identity component carries the connected theory developed here, while the finite component group acts by additional symmetries on the torus, roots, and representation ring. ## References - [Lie Groups I: Foundations](/page/Lie%20Groups%20I%3A%20Foundations) - [Lie Algebras I: Foundations](/page/Lie%20Algebras%20I%3A%20Foundations) - [Cambridge II Representation Theory](/page/Cambridge%20II%20Representation%20Theory) - [Hilbert Space](/page/Hilbert%20Space) - [Existence and Uniqueness of Haar Measure](/theorems/1063) - [Chambers and Positive Systems of a Root System](/theorems/4698) - [Coxeter Presentation of the Weyl Group](/theorems/4707)