example of matrix representations

Sign representation of Sn
Let G=Sn the n-th symmetric groupMathworldPlanetmathPlanetmath, and consider X(σ)=sign(σ) where σ is any permutationMathworldPlanetmath in Sn. That is, sign(σ)=1 when σ is an even permutationMathworldPlanetmath, and sign(σ)=-1 when σ is an odd permutation.

The function X is a group homomorphismMathworldPlanetmath between Sn and GL()={0} (that is invertible matrices of size 1×1, which is the set of non-zero complex numbersPlanetmathPlanetmath). And thus we say that {0} carries a representation of the symmetric group.

Defining representation of Sn
For each σSn, let X:SnGLn() the function given by X(σ)=(aij)n×n where (aij) is the permutation matrixMathworldPlanetmath given by

aij={1if σ(i)=j0if σ(i)j

Such matrices are called permutation matrices because they are obtained permuting the colums of the identity matrixMathworldPlanetmath. The function so defined is then a group homomorphism, and thus GLn() carries a representation of the symmetric group.

Title example of matrix representationsPlanetmathPlanetmath
Canonical name ExampleOfMatrixRepresentations
Date of creation 2013-03-22 14:53:31
Last modified on 2013-03-22 14:53:31
Owner drini (3)
Last modified by drini (3)
Numerical id 6
Author drini (3)
Entry type Example
Classification msc 20C99