signature of a permutation

Let $X$ be a finite set, and let $G$ be the group of permutations of $X$ (see permutation group). There exists a unique homomorphism $\chi$ from $G$ to the multiplicative group $\{-1,1\}$ such that $\chi(t)=-1$ for any transposition (loc. sit.) $t\in G$ . The value $\chi(g)$ , for any $g\in G$ , is called the signature or sign of the permutation $g$ . If $\chi(g)=1$ , $g$ is said to be of even parity; if $\chi(g)=-1$ , $g$ is said to be of odd parity.

Proposition: If $X$ is totally ordered by a relation $<$ , then for all $g\in G$ , \begin{equation} \chi(g)=(-1)^{k(g)} \end{equation}where $k(g)$ is the number of pairs $(x,y)\in X\times X$ such that $x<y$ and $g(x)>g(y)$ . (Such a pair is sometimes called an inversion of the permutation $g$ .)

Proof: This is clear if $g$ is the identity map $X\to X$ . If $g$ is any other permutation, then for some consecutive $a,b\in X$ we have $a<b$ and $g(a)>g(b)$ . Let $h\in G$ be the transposition of $a$ and $b$ . We have \begin{eqnarray*} k(g \circ h)&=&k(g)-1 \\ \chi(g \circ h)&=&-\chi(g) \end{eqnarray*}and the proposition follows by induction on $k(g)$ .