which maps to the corresponding permutation matrix (here is an arbitrary field). 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 .
as only 3 is fixed
as no elements of are fixed, and
as every element of is fixed.
^Dixon, John D.; Mortimer, Brian (2012-12-06). Permutation Groups. Springer Science & Business Media. pp. 5–6. ISBN 9781461207313.
^Robinson, Derek J. S. (2012-12-06). A Course in the Theory of Groups. Springer Science & Business Media. ISBN 9781468401288.