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Permutation representation

## Summary

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

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

The image ${\displaystyle \rho (G)\subset \operatorname {Sym} (X)}$  is a permutation group and the elements of ${\displaystyle G}$  are represented as permutations of ${\displaystyle X}$ .[1] A permutation representation is equivalent to an action of ${\displaystyle G}$  on the set ${\displaystyle X}$ :

${\displaystyle G\times X\to X.}$

See the article on group action for further details.

## Linear permutation representation

If ${\displaystyle G}$  is a permutation group of degree ${\displaystyle n}$ , then the permutation representation of ${\displaystyle G}$  is the linear representation of ${\displaystyle G}$

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

which maps ${\displaystyle g\in G}$  to the corresponding permutation matrix (here ${\displaystyle K}$  is an arbitrary field).[2] That is, ${\displaystyle G}$  acts on ${\displaystyle K^{n}}$  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 ${\displaystyle G}$  as a group of permutation matrices. One first represents ${\displaystyle G}$  as a permutation group and then maps each permutation to the corresponding matrix. Representing ${\displaystyle G}$  as a permutation group acting on itself by translation, one obtains the regular representation.

## Character of the permutation representation

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

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

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

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