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.

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.