This course develops the algebraic theory of symmetric functions, beginning with symmetric polynomials in finitely many variables and then passing to the stable ring that captures their behavior as the number of variables grows. The central goal is to understand the structure of this ring, its standard bases, and the algebraic operations and identities that organize the subject. Along the way, the course connects formal power series, representation-theoretic ideas, and enumerative combinatorics in a single framework.

The early chapters build the foundational language: symmetric polynomials, classical bases, inner products, involutions, and duality. From there, the course moves to Schur functions via Young tableaux and determinantal formulae, then studies the Cauchy identities and kernel methods that explain why these functions are so central. Later chapters extend these tools to Pieri rules, skew Schur functions, and the Littlewood-Richardson rule, which together govern multiplication and decomposition phenomena.

The final part of the course explores specializations, generating operators, and transition coefficients, using the theory to derive enumerative consequences and positivity results. The chapters are arranged so that each new idea refines or generalizes the previous ones: basic algebraic structure leads to explicit bases, which lead to identities, rules, and combinatorial formulas, and ultimately to a deeper understanding of how symmetric functions encode rich and pervasive positivity phenomena.

# Introduction

This course develops symmetric functions as a common language for enumeration, polynomial identities, and tableau combinatorics. The guiding question is how a symmetric polynomial identity in finitely many variables can be made stable as the number of variables grows, and how that stable object remembers enough structure to support explicit calculation. The main objects are the classical bases of the ring of symmetric functions, especially the monomial, elementary, complete homogeneous, power sum, and Schur bases.

The course is algebraic and combinatorial in emphasis. Representation theory explains why many of the identities are natural: Schur functions later match irreducible polynomial representations of general linear groups, the Hall inner product reflects character orthogonality, and the Littlewood-Richardson coefficients also govern tensor-product multiplicities. These notes keep the proofs inside polynomial algebra, generating functions, determinants, tableaux, and sign-reversing involutions, while pointing out where those outside interpretations are being shadowed. By the end of the course, the Littlewood-Richardson coefficients will appear as structure constants for Schur functions and as counts of tableaux satisfying a precise word condition.

## What Problem Symmetric Functions Solve

A polynomial such as

\begin{align*} x_1x_2+x_1x_3+x_2x_3 \end{align*}

is unchanged when the variables are permuted. This symmetry appears in many enumeration problems, but the number of variables often has no intrinsic meaning: it may be a temporary bound on the size of an alphabet, the number of possible colours, or the number of roots of an auxiliary polynomial. The first problem is to separate the stable algebraic content from the finite-variable presentation.

Throughout the finite-variable discussion, $k[x_1,\dots,x_n]$ denotes the polynomial ring in the variables $x_1,\dots,x_n$ with coefficients in the field or commutative ring $k$. The symmetric group $S_n$ acts on this ring by permuting the variables, and $k[x_1,\dots,x_n]^{S_n}$ denotes the subring of polynomials fixed by every such permutation.

[definition: Symmetric Polynomial] Let $k$ be a field and let $n \in \mathbb N$. A polynomial $f \in k[x_1, \dots, x_n]$ is symmetric if \begin{align*} f(x_{\sigma(1)}, \dots, x_{\sigma(n)}) = f(x_1, \dots, x_n) \end{align*} for every $\sigma \in S_n$. [/definition]

This definition is finite-variable, but the course quickly asks which statements survive when $n$ changes. The elementary symmetric polynomials give the first answer, so they must be named before the structure theorem can be stated.

[definition: Elementary Symmetric Polynomial] Let $k$ be a field and let $n \in \mathbb N$. For $0 \le r \le n$, the $r$th elementary symmetric polynomial in $k[x_1, \dots, x_n]$ is \begin{align*} e_r(x_1, \dots, x_n)=\\sum_{1 \le i_1<\cdots<i_r \le n}x_{i_1}\cdots x_{i_r}, \end{align*} with $e_0=1$. [/definition]

These polynomials encode square-free choices of variables. To turn symmetry into a workable algebraic condition, we need to know whether every symmetric polynomial can be reconstructed from these square-free coordinate functions. The finite-variable theorem says that they are not merely examples of symmetric polynomials: they are coordinates for all symmetric polynomials in $n$ variables.

[quotetheorem:5179]

[citeproof:5179]

The hypothesis that $f$ be symmetric is essential: $x_1$ cannot be expressed as a polynomial in $e_1, \dots, e_n$ when $n>1$, because every such expression is fixed by all permutations of the variables. The theorem also does not say that the same finite list of generators works uniformly for all $n$; the generator $e_n$ disappears after setting $x_n=0$. This limitation is the reason the course passes from symmetric polynomials in a fixed number of variables to the stable ring of symmetric functions.

[example: Elementary Symmetric Polynomials In Three Variables] In $k[x_1,x_2,x_3]$, the elementary symmetric polynomials are \begin{align*} e_1 &= x_1+x_2+x_3,\\ e_2 &= x_1x_2+x_1x_3+x_2x_3,\\ e_3 &= x_1x_2x_3. \end{align*} The symmetric polynomial $x_1^2+x_2^2+x_3^2$ is not one of these three elementary generators, but it can be written in terms of them. Expanding $e_1^2$ gives \begin{align*} e_1^2 &=(x_1+x_2+x_3)^2\\ &=x_1^2+x_1x_2+x_1x_3+x_2x_1+x_2^2+x_2x_3+x_3x_1+x_3x_2+x_3^2\\ &=x_1^2+x_2^2+x_3^2+2x_1x_2+2x_1x_3+2x_2x_3\\ &=x_1^2+x_2^2+x_3^2+2e_2. \end{align*} Subtracting $2e_2$ from both sides gives \begin{align*} e_1^2-2e_2 &=x_1^2+x_2^2+x_3^2+2e_2-2e_2\\ &=x_1^2+x_2^2+x_3^2. \end{align*} Thus \begin{align*} x_1^2+x_2^2+x_3^2=e_1^2-2e_2. \end{align*} This example shows how a symmetric polynomial can be converted into coordinates built from the elementary symmetric polynomials. [/example]

## The Stable Ring And Its Bases