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 $g\in G$,

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