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 [James and Kerber, 1981] is used. If Al is set to "Boerner" the method described in the book of Boerner [Boerner, 1967] 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 [Specht, 1935].

Example: integral representations (ex-be2f6e)

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

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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 [James and Kerber, 1981, 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: Semi Orthog (ex-5248de)

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

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They should both be of finite order, \(3\).

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

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