signature of a permutation

Let X be a finite setMathworldPlanetmath, and let G be the group of permutationsMathworldPlanetmath of X (see permutation groupMathworldPlanetmath). There exists a unique homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath χ from G to the multiplicative groupMathworldPlanetmath {-1,1} such that χ(t)=-1 for any transpositionMathworldPlanetmath (loc. sit.) tG. The value χ(g), for any gG, is called the signaturePlanetmathPlanetmathPlanetmath or sign of the permutation g. If χ(g)=1, g is said to be of even parity; if χ(g)=-1, g is said to be of odd parity.

PropositionPlanetmathPlanetmathPlanetmath: If X is totally ordered by a relationMathworldPlanetmathPlanetmath <, then for all gG,

χ(g)=(-1)k(g) (1)

where k(g) is the number of pairs (x,y)X×X such that x<y and g(x)>g(y). (Such a pair is sometimes called an inversionMathworldPlanetmath of the permutation g.)

Proof: This is clear if g is the identity mapMathworldPlanetmath XX. If g is any other permutation, then for some consecutive a,bX we have a<b and g(a)>g(b). Let hG be the transposition of a and b. We have

k(gh) = k(g)-1
χ(gh) = -χ(g)

and the proposition follows by inductionMathworldPlanetmath 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