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signature of a permutation

this entry is about the signature of a permutation. For other notions of signature see there.

Definition

For $\mathrm{Aut}(\{1,\cdots ,n\})\simeq S_n$ the symmetric group on $n\in \mathbb{N}$ elements, the signature is the unique group homomorphism

$$\mathrm{sign}:S_n\to {\mathbb{Z}}_2=\{1,-1\}$$sign : S_n \to \mathbb{Z}_2 = \{1, -1\}

that sends each transposition $s_{i,i+1}:\{1,\cdots ,n\}\to \{1,\cdots ,n\}$, which interchanges the $i$th element with its neighbour and leaves the other elements fixed, to the nontrivial element $(-1)\in {\mathbb{Z}}_2$.

Permutations in the kernel of $\mathrm{sign}$ are called even permutations, and the rest are called odd permutations.

Lemma

The signature is well-defined.

Proof

One way of seeing this is invoking a standard group presentation of $S_n$ where generators ${\sigma }_i$ for $i=1$ to $n-1$ (representing $s_{i,i+1}$) are subject to relations

$${\sigma }_i^2=1,({\sigma }_i{\sigma }_{i+1}{)}^3=1,{\sigma }_i{\sigma }_j={\sigma }_j{\sigma }_i(\mid i-j\mid >1),$$\sigma_{i}^2 = 1, \qquad (\sigma_i \sigma_{i+1})^3 = 1, \qquad \sigma_i \sigma_j = \sigma_j \sigma_i \; ({|i-j|} \gt 1),

and checking that the sign applied to both sides of a relation equation gives the same result.

Another is by invoking a tautological representation of $S_n$ on a polynomial algebra $\mathbb{Z}[x_1,\dots ,x_n]$,

$$S_n\stackrel{\cong }{\to }\mathrm{Set}(\{x_1,\dots ,x_n\},\{x_1,\dots ,x_n\})\to \mathrm{CRing}(\mathbb{Z}[x_1,\dots ,x_n],\mathbb{Z}[x_1,\dots ,x_n])$$S_n \stackrel{\cong}{\to} Set(\{x_1, \ldots, x_n\}, \{x_1, \ldots, x_n\}) \to CRing(\mathbb{Z}[x_1, \ldots, x_n], \mathbb{Z}[x_1, \ldots, x_n])

and recognizing that for the special polynomial

$$D≔\prod _{i<j}(x_i-x_j)$$D \coloneqq \prod_{i \lt j} (x_i - x_j)

we have, for each permutation $\tau$, either $\tau \cdot D=D$ or $\tau \cdot D=-D$. (The polynomial $\Delta ≔D^2$, which is invariant under the action, is called the discriminant?.)

Computations

There are various means for computing the signature (also called sign) of a permutation.

The definition itself suggests one method: if we linearly order the set $\{x_1,\dots ,x_n\}$ by $x_1<\dots <x_n$, then we can exhibit a permutation $\tau$ by a string diagram and simply count the number of crossings $I(\tau )$; then we have

$$\mathrm{sign}(\tau )=(-1{)}^{I(\tau )}.$$sign(\tau) = (-1)^{I(\tau)}.

Each crossing corresponds to a pair of elements $x_i<x_j$ such that $\tau (x_i)>\tau (x_j)$, called an inversion.

Another method which does not depend on choosing a total order is to exhibit a permutation through its cycle decomposition. Each cycle of period $k$ contributes a sign $(-1{)}^{k-1}$, and the overall sign is the product of these contributions taken over all the cycles.