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 representationEdit

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 representationEdit

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 representationEdit

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

  as only 3 is fixed
  as no elements of   are fixed, and
  as every element of   is fixed.

ReferencesEdit

  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.

External linksEdit