Characters and Representations

A set of functions are provided for computing with the characters and representations of a group. A full account of the character functions may be found in Chapter CHARACTERS OF FINITE GROUPS. Full details of the functions for constructing and analyzing representations may be found in Chapter MODULES OVER AN ALGEBRA AND GROUP REPRESENTATIONS. For the reader's convenience we include here a description of the basic functions for creating characters and representations.

Some functions described in this section may be missing or may have slightly different calling sequences for some categories of groups. For a complete description of the features available for a special category of groups, we refer to the chapter devoted to that category.

Contents

Character Theory

CharacterDegrees(G) : GrpPC -> [ Tup ]
Given a finite pc-group G, return the sequence [< d1, c1 >, < d2, c2 >, ... ], where ci is the number of irreducible characters of G having degree di. For details of the algorithm see Conlon [Con90].
CharacterTable(G) : GrpFin -> TabChtr
Construct the table of irreducible characters for the group G.
PermutationCharacter(G) : GrpPerm -> AlgChtrElt
Given a group G represented as a permutation group, construct the character of G afforded by the defining permutation representation of G.
PermutationCharacter(G, H) : GrpFin, GrpFin -> AlgChtrElt
Given a group G and some subgroup H of G, construct the ordinary character of G afforded by the permutation representation of G given by the action of G on the coset space of the subgroup H in G.
BurnsideCokernel(G) : Grp -> GrpAb, UserProgram, SeqEnum[AlgChtrElt]
For a finite group G computes the Burnside cokernel C(G), the quotient of the rational representation ring RQ(G) by the image of the Burnside ring (virtual permutation representations). Returns C(G) as a finite abelian group, map RQ(G)to C(G), and a list of irreducible rational characters with non-trivial class in C(G). (The implementation follows [AD14], and does not compute the subgroup lattice.)

Example Grp_grp-burnsidecokernel (H63E29)

The first example of a finite group with C(G)≠1 is due to Serre, G=C3 x Q8. In fact, there are two groups of order 24 with C(G)≠1 and no smaller examples: 
> [GroupName(G): G in SmallGroups(n), n in [1..24] | #BurnsideCokernel(G) ne 1];
[ C3:C8, C3*Q8 ]

Representation Theory

We describe the main functions for creating K[G]-modules for finite groups. The machinery for working with these modules is described in Chapter MODULES OVER AN ALGEBRA AND GROUP REPRESENTATIONS.

GModule(G, S) : GrpFin, AlgMat -> ModGrpFin
Let G be a group defined on r generators and let S be a subalgebra of the matrix algebra Mn(R), also defined by r non-singular matrices. It is assumed that the mapping from G to S defined by φ : G.i |-> S.i, for i = 1, ..., r, extends to a group homomorphism. Let M be the natural module for the matrix algebra S. The function GModule gives M the structure of an S[G]-module, where the action of the i-th generator of G on M is given by the i-th generator of S.
GModule(G, A, B) : GrpFin, GrpFin, GrpFin -> ModGrpFin, Map
Given a finite group G, a normal subgroup A of G and a normal subgroup B of A such that the section A/B is elementary abelian of order pn, create the K[G]-module M corresponding to the action of G on A/B, where K is the field GF(p). If B is trivial, it may be omitted. The function returns:
(a)
the module M; and,
(b)
the homomorphism φ : A/B -> M.
PermutationModule(G, H, R) : GrpFin, GrpFin, Rng -> ModGrpFin
Given a finite group G and a ring R, create the R[G]-module for G corresponding to the permutation action of G on the cosets of H.
PermutationModule(G, R) : GrpPerm, Rng -> ModGrpFin
Given a finite permutation group G and a ring R, create the natural permutation module for G over R.

Example Grp_Modules (H63E30)

The permutation module for the group M10 over GF(2) may be created as follows: 
> m10 := PermutationGroup< 10 |  (1, 3, 9, 10, 2, 8, 7, 6, 4, 5),
>                                (1, 7)(2, 4, 3, 6, 8, 10, 9, 5) >;
> p := PermutationModule(m10, GF(2));
> p : Maximal;
GModule p of dimension 10 over GF(2)
Generators of acting algebra:
[0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 0]
[1 0 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0]
[1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0]

Example Grp_Modules-2 (H63E31)

The group G defined below is the split extension of an elementary abelian group E of order 16 by Alt(6). After setting up the group, we construct the module M for G corresponding to its action on E. 
> G := PermutationGroup< 16 |
>         (1, 15, 7, 5, 12)(2, 9, 13, 14, 8)(3, 6, 10, 11, 4),
>         (1, 4, 5)(2, 8, 10)(3, 12, 15)(6, 13, 11)(7, 9, 14),
>         (1, 16)(2, 3)(4, 5)(6, 7)(8, 9)(10, 11)(12, 13)(14, 15) >;
> CS := ChiefSeries(G);
> [ Order(H) : H in CS ];
[ 5760, 16, 1 ]
> M := GModule(G, CS[2]);
> M:Maximal;
GModule M of dimension 4 over GF(2)
Generators of acting algebra:
[0 1 0 0]
[0 1 1 0]
[0 0 1 1]
[1 0 0 1]
[0 0 1 0]
[0 0 0 1]
[1 0 1 0]
[0 1 0 1]
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
V2.28, 13 July 2023