mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

s, or as a group of

permutation matrices In mathematics, particularly in Matrix (mathematics), matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. An permutation matrix can represent a permu ...

. The term also refers to the combination of the two.

Abstract permutation representation

A permutation representation of a group

G

on a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

X

is a

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

from

G

to the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

X

: :

\rho\colon G \to \operatorname(X).

The image

\rho(G)\sub \operatorname(X)

is a

permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...

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 In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...

for further details.

Linear permutation representation

G

is a

of degree

n

, then the permutation representation of

G

is the linear representation of

G

\rho\colon G\to \operatorname_n(K)

which maps

g\in G

to the corresponding

permutation matrix In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. An permutation matrix can represent a permutation of elements. ...

(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 Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...

, 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)

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=\

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(\begin 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\end)=1

as only 3 is fixed :

\chi((123))=\operatorname(\begin 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end)=0

as no elements of

X

are fixed, and :

\chi(1)=\operatorname(\begin 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end)=3

as every element of

X

is fixed.

References

Representation theory of finite groups Permutation groups

External links

*https://mathoverflow.net/questions/286393/how-do-i-know-if-an-irreducible-representation-is-a-permutation-representation {{Abstract-algebra-stub