signature of a permutation

Let $X$ be a finite set, and let $G$ be the group of permutations of $X$ (seepermutation group). There exists a unique homomorphism $\chi$ from $G$ to themultiplicative 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 thesignature 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 bya 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 and $g(x)>g(y)$. (Such a pair is sometimes called an inversionof the permutation $g$.)

Proof: This is clear if $g$ is the identity map $X\to X$.If $g$ is any other permutation, then for someconsecutive $a,b\in X$ we have and $g(a)>g(b)$. Let $h\in G$be the transposition of $a$ and $b$. We have

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

$\chi (g\circ h)$$=$$-\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