Inmathematics, the termpermutation representation of a (typically finite)group${\displaystyle G}$ can refer to either of two closely related notions: arepresentation of${\displaystyle G}$ as a group ofpermutations, or as a group ofpermutation matrices. The term also refers to the combination of the two.

Apermutation representation of agroup${\displaystyle G}$ on aset${\displaystyle X}$ is ahomomorphism from${\displaystyle G}$ to thesymmetric group of${\displaystyle X}$:

${\displaystyle \rho :G\to \mathrm{Sym}(X).}$

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

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

See the article ongroup action for further details.

If${\displaystyle G}$ is apermutation group of degree${\displaystyle n}$, then thepermutation representation of${\displaystyle G}$ is thelinear representation of${\displaystyle G}$

${\displaystyle \rho :G\to {\mathrm{GL}}_n(K)}$

which maps${\displaystyle g\in G}$ to the correspondingpermutation matrix (here${\displaystyle K}$ is an arbitraryfield).^[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 bytranslation, one obtains theregular representation.

Given a group${\displaystyle G}$ and a finite set${\displaystyle X}$ with${\displaystyle G}$ acting on the set${\displaystyle X}$ then thecharacter${\displaystyle \chi }$ 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

as only 3 is fixed

as no elements of${\displaystyle X}$ are fixed, and

as every element of${\displaystyle X}$ is fixed.

• https://mathoverflow.net/questions/286393/how-do-i-know-if-an-irreducible-representation-is-a-permutation-representation

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