Permutation representation

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 (mathematics) ${\displaystyle \chi }$ 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.