Permutation representation

A permutation representation of a group $G$ on a set $X$ is a homomorphism from $G$ to the symmetric group of $X$:

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

The image $\backslash rho(G)\backslash sub\; \backslash operatorname\{Sym\}(X)$ is a permutation group and the elements of $G$ are represented as permutations of $X$.{{Cite book|url=https://books.google.com/books?id=1SPjBwAAQBAJ|title=Permutation Groups|last=Dixon|first=John D.|last2=Mortimer|first2=Brian|date=2012-12-06|publisher=Springer Science & Business Media|isbn=9781461207313|pages=5–6|language=en}} A permutation representation is equivalent to an action of $G$ on the set $X$:

:$G\backslash times\; X\; \backslash to\; X.$

See the article on group action for further details.

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

:$\backslash rho\backslash colon\; G\backslash to\; \backslash operatorname\{GL\}\_n(K)$

which maps $g\backslash in\; G$ to the corresponding permutation matrix (here $K$ is an arbitrary field).{{Cite book|url=https://books.google.com/books?id=BFrTBwAAQBAJ|title=A Course in the Theory of Groups|last=Robinson|first=Derek J. S.|date=2012-12-06|publisher=Springer Science & Business Media|isbn=9781468401288|language=en}} 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.

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

This follows since, if we represent the map $\backslash rho(g)$ 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 $X=\backslash \{1,\; 2,\; 3\backslash \}$ the character of the permutation representation can be computed with the formula $\backslash chi(g)=$ the number of points of $X$ fixed by $g$.

So

: as only 3 is fixed

: as no elements of $X$ are fixed, and

: as every element of $X$ is fixed.

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Category:Representation theory of finite groups

Category:Permutation groups

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

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