GAP:IrrepDegreeGroupingFull

Definition

Function type

This is a coded (not in-built) function in GAP to group together all groups of a given order based on their degrees of irreducible representations.

Behavior

The function is invoked as:

IrrepDegreeGroupingFull(n)

where $n$ is a natural number. The output is a list of pairs of lists, where each pair is as follows:

• The first member of the pair is itself a list of pairs describing a collection of degrees of irreducible representations over $\mathbb{C}$: the first member of each pair is a degree and the second member is the number of irreducible representations of that degree.
• The second member of the pair is the list of second parts of GAP IDs of groups of order $n$ having those degrees of irreducible representations.

Code

IrrepDegreeGroupingFull := function(n)
        local L,M,m,N;
        L := List(AllSmallGroups(n),CharacterDegrees);
        M := SortedList(Set(L));
        m := Length(L);
        N := List(M,i -> [i,Filtered([1..m],j->L[j]=i)]);
        return(SortedList(N));
end;;

Examples of usage

$n$	Function call	Output	Interpretation
1	IrrepDegreeGroupingFull(1)	[ [ [ [ 1, 1 ] ], [ 1 ] ] ]	The group with GAP ID $(1,1)$ has 1 irreducible representation of degree 1.
8	IrrepDegreeGroupingFull(8)	[ [ [ [ 1, 4 ], [ 2, 1 ] ], [ 3, 4 ] ], [ [ [ 1, 8 ] ], [ 1, 2, 5 ] ] ]	The groups with GAP IDs $(8,3)$ and $(8,4)$ both have four irreducible representations of degree 1 and one irreducible representation of degree 2. The groups with GAP IDs $(8,1)$, $(8,2)$, and $(8,5)$ each have 8 irreducible representations of degree 1.