Permutation representation

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

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 .[1] 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 (here is an arbitrary field).[2] 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 (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.

References

  1. Dixon, John D.; Mortimer, Brian (2012-12-06). Permutation Groups. Springer Science & Business Media. pp. 5–6. ISBN 9781461207313.
  2. Robinson, Derek J. S. (2012-12-06). A Course in the Theory of Groups. Springer Science & Business Media. ISBN 9781468401288.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.