GAP:CharacterDegrees

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

Related functions

 * GAP:CharacterTable

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)[2],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 ] ]