GAP:IrrepDegreeGroupingFull

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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.