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Representations of the Symmetric Group

For the symmetric group of degree n the irreducible representations can be indexed by partitions of weight n. For more information on partitions see Section Partitions.

Integral Representations

It is possible to define representing matrices of the symmetric group over the integers.

SymmetricRepresentation(pa, pe) : SeqEnum, GrpPermElt -> AlgMatElt

    Al: MonStgElt                       Default: "JamesKerber"

Given a partition pa of weight n and a permutation pe in a symmetric group of degree n, return an irreducible representing matrix for pe, indexed by pa, over the integers. If Al is set to the default "JamesKerber" then the method described in [JK81] is used. If Al is set to "Boerner" the method described in the book of Boerner [Boe67] is used. If Al is set to "Specht" then the method used is a direct implementation of that used by Specht in his paper from 1935 [Spe35].

Example `RepSym_integral representations (H99E1)`

We compute a representing matrix of a permutation using two different algorithms and check whether the results have the same character.

> a:=SymmetricRepresentation([3,2],Sym(5)!(3,4,5) : Al := "Boerner");a;
[ 0  0  1 -1  0]
[ 1  0  0 -1  0]
[ 0  1  0 -1  0]
[ 0  0  0 -1  1]
[ 0  0  0 -1  0]
> b:=SymmetricRepresentation([3,2],Sym(5)!(3,4,5) : Al := "Specht");b;
[ 0  1  0 -1  0]
[ 0  0  1  0 -1]
[ 1  0  0  0  0]
[ 0  0  0  0 -1]
[ 0  0  0  1 -1]
> IsSimilar(Matrix(Rationals(), a), Matrix(Rationals(), b));
true

The matrices are similar as they should be.

The Seminormal and Orthogonal Representations

The seminormal and orthogonal representations involve matrices which are not necessarily integral. The method Magma uses to construct these matrices is described in [JK81, Section 3.3];

SymmetricRepresentationSeminormal(pa, pe) : SeqEnum,GrpPermElt -> AlgMatElt

Given a partition pa of weight n and a permutation pe in a symmetric group of degree n, return the matrix of the seminormal representation for pe, indexed by pa, over the rationals.

SymmetricRepresentationOrthogonal(pa, pe) : SeqEnum,GrpPermElt -> AlgMatElt

Given a partition pa of weight n and a permutation pe in a symmetric group of degree n, return the matrix of the orthogonal representation for pe, indexed by pa. An orthogonal basis is used to compute the matrix which may have entries in a cyclotomic field.

Example `RepSym_semi-orthog (H99E2)`

We compare the seminormal and orthogonal representations of a permutation and note that they are similar.

> g:=Sym(5)!(3,4,5);
> a:=SymmetricRepresentationSeminormal([3,2],g);a;
[-1/2    0 -3/4    0    0]
[   0  1/2    0  3/4    0]
[   1    0 -1/2    0    0]
[   0  1/3    0 -1/6  8/9]
[   0    1    0 -1/2 -1/3]
> b:=SymmetricRepresentationOrthogonal([3,2],g);b;
[-1/2 0 zeta(24)_8^2*zeta(24)_3 + 1/2*zeta(24)_8^2 0 0]
[0 1/2 0 -zeta(24)_8^2*zeta(24)_3 - 1/2*zeta(24)_8^2 0]
[-zeta(24)_8^2*zeta(24)_3 - 1/2*zeta(24)_8^2 0 -1/2 0 0]
[0 -1/3*zeta(24)_8^2*zeta(24)_3 - 1/6*zeta(24)_8^2 0
 -1/6 2/3*zeta(24)_8^3 - 2/3*zeta(24)_8]
[0 2/3*zeta(24)_8^3*zeta(24)_3 + 1/3*zeta(24)_8^3
+ 2/3*zeta(24)_8*zeta(24)_3 + 1/3*zeta(24)_8 0
    -1/3*zeta(24)_8^3 + 1/3*zeta(24)_8 -1/3]
> IsSimilar(a,b);
true

They should both be of finite order, 3.

> IsOne(a^Order(g));
true
> IsOne(b^Order(g));
true
>

V2.28, 13 July 2023

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