In mathematics, the term permutation representation of a (typically finite) 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
See the article on group action for further details.
Linear permutation representation
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 (mathematics) 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
- as only 3 is fixed
- as no elements of are fixed, and
- as every element of is fixed.