Permutation representation


In mathematics, the term permutation representation of a group can refer to either of two closely related notions: a representation of as a group of permutations, or as a group of permutation matrices. The term also refers to the combination of the two.

Abstract permutation representation

A permutation representation of a group on a set is a homomorphism from to the symmetric group of :
The image is a permutation group and the elements of are represented as permutations of. A permutation representation is equivalent to an action of on the set :
See the article on group action for further details.

Linear permutation representation

If is a permutation group of degree, then the permutation representation of is the linear representation of
which maps to the corresponding permutation matrix. That is, acts on by permuting the standard basis vectors.
This notion of a permutation representation can, of course, be composed with the previous one to represent an arbitrary abstract group as a group of permutation matrices. One first represents as a permutation group and then maps each permutation to the corresponding matrix. Representing as a permutation group acting on itself by translation, one obtains the regular representation.

Character of the permutation representation

Given a group and a finite set with acting on the set then the character of the permutation representation is exactly the number of fixed points of under the action of on. That is the number of points of fixed by.
This follows since, if we represent the map with a matrix with basis defined by the elements of we get a permutation matrix of. Now the character of this representation is defined as the trace of this permutation matrix. An element on the diagonal of a permutation matrix is 1 if the point in is fixed, and 0 otherwise. So we can conclude that the trace of the permutation matrix is exactly equal to the number of fixed points of.
For example, if and the character of the permutation representation can be computed with the formula the number of points of fixed by.
So