# GAP:CharacterDegrees

## Definition

The function has different input types, and the behavior is indicated below:

Input type Output type How to interpret the output
A single finite group A list of pairs of positive integers The list gives the degrees of irreducible representations of the finite group over a splitting field of characteristic zero. It is interpreted as follows: for each pair of positive integers, the first member is a degree of irreducible representation, and the second member is the number of irreducible representations of that degree.
A single finite group and a single number, which must be either zero or a prime A list of pairs of positive integers The list gives the degrees of irreducible representations of the finite group over a splitting field of characteristic equal to the second input. It is interpreted as follows: for each pair of positive integers, the first member is a degree of irreducible representation, and the second member is the number of irreducible representations of that degree.
A character table or Brauer character table A list of pairs of positive integers Gives in the list of pairs form the degrees of characters/Brauer characters occurring in the table, similar to above.

## Examples of usage

### For an input group without a specified characteristic

To find the degrees of irreducible representations for a finite group over a splitting field of characteristic zero:

```gap> CharacterDegrees(SymmetricGroup(4));
[ [ 1, 2 ], [ 2, 1 ], [ 3, 2 ] ]```

This is to be understood as follows: the group symmetric group:S4 has 2 irreducible representations of degree 1, 1 irreducible representation of degree 2, and 2 irreducible representations of degree 3.

Similarly:

```gap> CharacterDegrees(SmallGroup(24,3));
[ [ 1, 3 ], [ 2, 3 ], [ 3, 1 ] ]```

This means that the group with GAP ID (24,3) (see SmallGroup) has 3 irreducible representations of degree 1, 3 irreducible representations of degree 2, and 1 irreducible representation of degree 3. (The group is actually special linear group:SL(2,3) and its linear representation theory can be found at linear representation theory of special linear group:SL(2,3)).

```gap> List(AllSmallGroups(8),G -> [IdGroup(G),CharacterDegrees(G)]);
[ [ 1, [ [ 1, 8 ] ] ], [ 2, [ [ 1, 8 ] ] ], [ 3, [ [ 1, 4 ], [ 2, 1 ] ] ],
[ 4, [ [ 1, 4 ], [ 2, 1 ] ] ], [ 5, [ [ 1, 8 ] ] ] ]```

The above code lists, for all the groups of order 8, the degrees of irreducible representations of the group, labeled by its GAP ID (see linear representation theory of groups of order 8). The [1,1,8] indicates that the group with GAP ID (8,1) has 8 degree 1 irreducible representations. The [2,1,8] indicates that the group with GAP ID (8,2) also has 8 degree 1 irreducible representations. The [3,[[1,4],[2,1]]] indicates that the group with GAP ID (8,3) has 4 degree 1 irreducible representations and 1 degree 2 irreducible representation. And so on.

### For an input group with a specified characteristic

Here are some examples:

```gap> CharacterDegrees(SymmetricGroup(3),2);
[ [ 1, 1 ], [ 2, 1 ] ]
gap> CharacterDegrees(SymmetricGroup(4),2);
[ [ 1, 1 ], [ 2, 1 ] ]
gap> CharacterDegrees(SymmetricGroup(3),3);
[ [ 1, 2 ] ]
gap> CharacterDegrees(SymmetricGroup(4),3);
[ [ 1, 2 ], [ 3, 2 ] ]```