mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the term permutation representation of a (typically finite)

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

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 is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...

s, or as a group of

permutation matrices 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 and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when ...

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

Abstract permutation representation

A permutation representation of a

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 structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

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 group \m ...

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 it ...

and the elements of

G

are represented as permutations of

X

. A permutation representation is equivalent to an

action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...

G

on the set

X

: :

G\times X \to X.

See the article on

group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

for further details.

Linear permutation representation

G

is a

of degree

n

, then the permutation representation of

G

is the

linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...

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 and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when ...

(here

K

is an arbitrary

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

). 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 Meaning (linguistic), 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 ...

, one obtains the

regular representation In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation. One distinguishes the left regular rep ...

Character of the permutation representation

Given a group

G

and a finite set

X

with

G

acting on the set

X

then the

character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...

\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