# Characters of the Symmetric Group

## Single Values

The method used to compute the value of a character on a permutation is the recursion formula of Murnaghan and Nakayama [[James and Kerber, 1981](../../references.md#cite-jake), p. 60], except when computing the value of a character on the identity permutation a formula for the dimension of the representation indexed by a partition is used, [[James and Kerber, 1981](../../references.md#cite-jake), p. 56].

### `SymmetricCharacterValue(pa, pe): SeqEnum, GrpPermElt -> RngElt`

Computes the value of the irreducible character of the symmetric group of degree $n$ indexed by the partition $pa$ of weight $n$ on the permutation $pe$. When computing the value of the character on the identity permutation, i.e. the degree of the character, the dimension (hook) formula is used on the partition $pa$.

## Irreducible Characters

### `SymmetricCharacter(pa): SeqEnum -> AlgChtrElt`

Return the character of the representation of the symmetric group of degree $n$ indexed by the partition $pa$ where $pa$ has weight $n$.

## Character Table

### `SymmetricCharacterTable(d): RngIntElt -> SeqEnum`

Return the character table of the symmetric group of degree $d$.