A set of functions are provided for computing with the characters of a group. Full details of these functions may be found in the online help nodes on characters. For convenience we include here two of the more useful character functions.

Also, functions are provided for computing with the modular representations of a group. Full details of these functions may be found in the online help nodes on R[G]-modules. For the reader's convenience we include here the functions which may be used to define a R[G]-module for a permutation group.

CharacterTable(G) : GrpPerm -> TabChtr

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.

PermutationCharacter(G, H) : GrpPerm, GrpPerm -> 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.

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, is 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) : Grp, Grp, Grp -> ModGrp, 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 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.

PermutationModule(G, R) : GrpPerm, Rng -> ModGrp

Given a finite permutation group G and a ring R, create the natural permutation module for G over R.