Permutation representation

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