signature of a permutation

this entry is about the

signatureof a permutation. For other notions of signature see there.

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

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

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 $sign$ are called *even* permutations, and the rest are called *odd* permutations.

The signature is well-defined.

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

$\sigma_{i}^2 = 1, \qquad (\sigma_i \sigma_{i+1})^3 = 1, \qquad \sigma_i \sigma_j = \sigma_j \sigma_i \; ({|i-j|} \gt 1),$

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 $\mathbb{Z}[x_1, \ldots, x_n]$,

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

and recognizing that for the special polynomial

$D \coloneqq \prod_{i \lt j} (x_i - x_j)$

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

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 $\{x_1, \ldots, x_n\}$ by $x_1 \lt \ldots \lt x_n$, then we can exhibit a permutation $\tau$ by a string diagram and simply count the number of crossings $I(\tau)$; then we have

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

Each crossing corresponds to a pair of elements $x_i \lt x_j$ such that $\tau(x_i) \gt \tau(x_j)$, 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.

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