# 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.

If $X$ is totally ordered by a relation $<$, then for all $g\in G$,

 $\chi(g)=(-1)^{k(g)}$ (1)

where $k(g)$ is the number of pairs $(x,y)\in X\times X$ such that $x 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 and $g(a)>g(b)$. Let $h\in G$ be the transposition of $a$ and $b$. We have

 $\displaystyle k(g\circ h)$ $\displaystyle=$ $\displaystyle k(g)-1$ $\displaystyle\chi(g\circ h)$ $\displaystyle=$ $\displaystyle-\chi(g)$

and the proposition follows by induction on $k(g)$.

 Title signature of a permutation Canonical name SignatureOfAPermutation Date of creation 2013-03-22 13:29:19 Last modified on 2013-03-22 13:29:19 Owner rspuzio (6075) Last modified by rspuzio (6075) Numerical id 9 Author rspuzio (6075) Entry type Definition Classification msc 03-00 Classification msc 05A05 Classification msc 20B99 Synonym sign of a permutation Related topic Transposition Defines inversion Defines signature Defines parity Defines even permutation Defines odd permutation