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## Summary

In mathematics, the term permutation representation of a (typically finite) group $G$ can refer to either of two closely related notions: a representation of $G$ 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 $G$  on a set $X$  is a homomorphism from $G$  to the symmetric group of $X$ :

$\rho \colon G\to \operatorname {Sym} (X).$

The image $\rho (G)\subset \operatorname {Sym} (X)$  is a permutation group and the elements of $G$  are represented as permutations of $X$ . A permutation representation is equivalent to an action of $G$  on the set $X$ :

$G\times X\to X.$

See the article on group action for further details.

## Linear permutation representation

If $G$  is a permutation group of degree $n$ , then the permutation representation of $G$  is the linear representation of $G$

$\rho \colon G\to \operatorname {GL} _{n}(K)$

which maps $g\in G$  to the corresponding permutation matrix (here $K$  is an arbitrary field). That is, $G$  acts on $K^{n}$  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 $G$  as a group of permutation matrices. One first represents $G$  as a permutation group and then maps each permutation to the corresponding matrix. Representing $G$  as a permutation group acting on itself by translation, one obtains the regular representation.

## Character of the permutation representation

Given a group $G$  and a finite set $X$  with $G$  acting on the set $X$  then the character $\chi$  of the permutation representation is exactly the number of fixed points of $X$  under the action of $\rho (g)$  on $X$ . That is $\chi (g)=$  the number of points of $X$  fixed by $\rho (g)$ .

This follows since, if we represent the map $\rho (g)$  with a matrix with basis defined by the elements of $X$  we get a permutation matrix of $X$ . 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 $X$  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 $X$ .

For example, if $G=S_{3}$  and $X=\{1,2,3\}$  the character of the permutation representation can be computed with the formula $\chi (g)=$  the number of points of $X$  fixed by $g$ . So

$\chi ((12))=\operatorname {tr} ({\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}})=1$  as only 3 is fixed
$\chi ((123))=\operatorname {tr} ({\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}})=0$  as no elements of $X$  are fixed, and
$\chi (1)=\operatorname {tr} ({\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}})=3$  as every element of $X$  is fixed.