A permutation of a finite set looks like a harmless reordering until a formula starts to care about orientation. The determinant changes sign when two rows are swapped, an alternating tensor changes sign when two inputs are exchanged, and the roots of a polynomial split into orderings whose parity can carry field-theoretic information. This places signature directly between the group theory of symmetric and alternating groups, the linear algebra of determinants, and the Galois-theoretic action on polynomial roots. The signature of a permutation is the invariant that records whether a reordering preserves or reverses this orientation.

The first problem is that a permutation has many descriptions. A cycle can be expanded into transpositions in more than one way, and extra cancelling swaps can be inserted without changing the permutation. If orientation is to be encoded by the number of swaps, then the parity of that number must be intrinsic rather than a feature of the chosen expression.

[example: A Row Swap Changes Orientation] Let $V=\mathbb{R}^2$ with ordered basis $e_1=(1,0)$ and $e_2=(0,1)$. Here $S_2$ denotes the symmetric group on $\{1,2\}$, and let $\sigma\in S_2$ be the transposition exchanging $1$ and $2$. The ordered pair $(e_1,e_2)$ is represented by the column data first column $(1,0)$ and second column $(0,1)$, so the $2\times 2$ determinant formula gives \begin{align*} \det(e_1,e_2)=1\cdot 1-0\cdot 0=1. \end{align*} After applying the swap, the ordered pair becomes $(e_2,e_1)$, whose first column is $(0,1)$ and whose second column is $(1,0)$. Hence \begin{align*} \det(e_2,e_1)=0\cdot 0-1\cdot 1=-1. \end{align*} For this permutation, the only pair with $i<j$ is $(1,2)$, and $\sigma(1)=2>\sigma(2)=1$, so $\operatorname{Inv}(\sigma)=\{(1,2)\}$. Therefore \begin{align*} \operatorname{sgn}(\sigma)=(-1)^{|\operatorname{Inv}(\sigma)|}=(-1)^1=-1. \end{align*} Thus the swap changes the determinant by the factor $-1$, exactly matching the signature and reversing the orientation of the ordered basis. [/example]

This example is small, but it contains the general phenomenon. Signature is not a measure of how far a permutation moves points. It measures whether the reordering reverses the alternating orientation seen by determinants, wedge products, and root-difference products.

## Definition

Throughout this page, $n\in\mathbb{N}$ means $n\ge 1$, so all symmetric groups under discussion act on $\{1,\dots,n\}$.

The page topic is the sign itself: a function from the symmetric group to the two possible orientation values. To define it without making arbitrary choices about transposition decompositions, we count order reversals directly.

[definition: Signature] Let $n\in\mathbb{N}$. The signature, or sign, is the map \begin{align*} \operatorname{sgn}:S_n&\to\{1,-1\} \end{align*} defined by \begin{align*} \operatorname{sgn}(\sigma)=(-1)^{|\{(i,j):1\le i<j\le n,\ \sigma(i)>\sigma(j)\}|}. \end{align*} [/definition]

A permutation with signature $1$ is called even, and a permutation with signature $-1$ is called odd. The definition uses the natural order on $\{1,\dots,n\}$, but its output will turn out to be an invariant of the group operation, not a fragile feature of the chosen notation.

The exponent in the formula is doing the real work, so it needs its own language. Counting moved points is not enough: the list $3,5,1,4,2$ moves every symbol, while other permutations can move many symbols and still have either parity. What matters is whether a pair has crossed, meaning that two entries which originally appeared in increasing order now appear in decreasing order. Naming such a crossing as an inversion gives us a local unit of orientation reversal before we collect all reversals into the signature.

[definition: Inversion of a Permutation] Let $n\in\mathbb{N}$ and let $\sigma\in S_n$. An inversion of $\sigma$ is a pair $(i,j)$ with $1\le i<j\le n$ such that $\sigma(i)>\sigma(j)$. [/definition]

A single inversion is only a local crossing. To make later computations readable, we collect all such crossings into one finite set.

[definition: Inversion Set] Let $n\in\mathbb{N}$. Here $\mathcal{P}(X)$ denotes the power set of a set $X$, meaning the set of all subsets of $X$. The inversion-set map is the function \begin{align*} \operatorname{Inv}:S_n&\to\mathcal{P}(\{(i,j):1\le i<j\le n\}) \end{align*} defined by \begin{align*} \operatorname{Inv}(\sigma)=\{(i,j):1\le i<j\le n,\ \sigma(i)>\sigma(j)\}. \end{align*} [/definition]

With this notation, the defining formula becomes

\begin{align*} \operatorname{sgn}(\sigma)=(-1)^{|\operatorname{Inv}(\sigma)|}. \end{align*}

To see why this parity deserves to be called orientation, we need the elementary swaps from which all permutations are built. These swaps are the test case for every sign rule.

[definition: Transposition] Let $n\in\mathbb{N}$. A transposition in $S_n$ is a permutation \begin{align*} \tau:\{1,\dots,n\}&\to\{1,\dots,n\} \end{align*} for which there exist distinct $a,b\in\{1,\dots,n\}$ such that $\tau(a)=b$, $\tau(b)=a$, and $\tau(k)=k$ for every $k\notin\{a,b\}$. [/definition]

A transposition is the smallest possible non-identity reordering, but its effect on inversions is not limited to the two labels it exchanges. For instance, swapping two non-adjacent labels can reverse their order and also reverse the order of labels lying between them. The important point is therefore not the exact number of affected pairs, but the parity of that number: a single swap should reverse orientation exactly once modulo $2$.

This creates the first local test for the inversion definition of sign. Before using transpositions as building blocks for arbitrary permutations, we must know that every individual transposition has the expected negative sign, even when the two exchanged positions are not adjacent.

The obstruction is that a distant swap changes many pairwise comparisons at once, so its sign is not immediate from the definition. The next result isolates exactly this issue: it checks that all those reversed comparisons still contribute an odd number of inversions, making every transposition orientation-reversing.