# nLab signature of a permutation

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

## Definition

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

$\mathrm{sign}:{S}_{n}\to {ℤ}_{2}=\left\{1,-1\right\}$sign : S_n \to \mathbb{Z}_2 = \{1, -1\}

that sends each transposition ${s}_{i,i+1}:\left\{1,\cdots ,n\right\}\to \left\{1,\cdots ,n\right\}$, which interchanges the $i$th element with its neighbour and leaves the other elements fixed, to the nontrivial element $\left(-1\right)\in {ℤ}_{2}$.

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

${\sigma }_{i}^{2}=1,\phantom{\rule{2em}{0ex}}\left({\sigma }_{i}{\sigma }_{i+1}{\right)}^{3}=1,\phantom{\rule{2em}{0ex}}{\sigma }_{i}{\sigma }_{j}={\sigma }_{j}{\sigma }_{i}\phantom{\rule{thickmathspace}{0ex}}\left(\mid i-j\mid >1\right),$\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 $ℤ\left[{x}_{1},\dots ,{x}_{n}\right]$,

${S}_{n}\stackrel{\cong }{\to }\mathrm{Set}\left(\left\{{x}_{1},\dots ,{x}_{n}\right\},\left\{{x}_{1},\dots ,{x}_{n}\right\}\right)\to \mathrm{CRing}\left(ℤ\left[{x}_{1},\dots ,{x}_{n}\right],ℤ\left[{x}_{1},\dots ,{x}_{n}\right]\right)$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 _{iD \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 $\left\{{x}_{1},\dots ,{x}_{n}\right\}$ by ${x}_{1}<\dots <{x}_{n}$, then we can exhibit a permutation $\tau$ by a string diagram and simply count the number of crossings $I\left(\tau \right)$; then we have

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

Each crossing corresponds to a pair of elements ${x}_{i}<{x}_{j}$ such that $\tau \left({x}_{i}\right)>\tau \left({x}_{j}\right)$, 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 $\left(-1{\right)}^{k-1}$, and the overall sign is the product of these contributions taken over all the cycles.

Revised on November 25, 2012 20:44:24 by Todd Trimble (67.81.93.16)