# 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](../../Combinatorics/PartitionsWordsTableaux/partitions.md#tableaux-partitions).

## Integral Representations

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

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

```magma
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](../../references.md#cite-jake)] is used. If `Al` is set to `"Boerner"` the method described in the book of Boerner [[Boerner, 1967](../../references.md#cite-boe2)] 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](../../references.md#cite-sp35)].

### `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.

```magma
> 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 [[James and Kerber, 1981](../../references.md#cite-jake), 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.

```magma
> 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, $3$.

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

```