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 A MATRIX ALGEBRA. 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.
Construct the table of irreducible characters for the group G. The characters are found using the Dixon-Schneider algorithm.
Given a group G represented as a permutation group, construct the character of G afforded by the defining permutation representation of G.
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.
Given a finite p-group G, return the sequence [C_0, C_1, ... ], where C_i is the number of irreducible characters of G having degree p^i.
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 A MATRIX ALGEBRA.
Let G be a group defined on r generators and let S be a subalgebra of the matrix algebra M_n(R), also defined by r non-singular matrices. It is assumed that the mapping from G to S defined by phi : 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.
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 p^n, 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:
- the module M; and,
- the homomorphism phi : A/B -> M.
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.
Given a finite permutation group G and a ring R, create the natural permutation module for G over R.
> 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]
> 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]