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 KG-modules. For the reader's convenience we include here the functions which may be used to define a KG-module for a matrix group.

LinearCharacters(G) : GrpMat -> [ Chtr ]

A sequence containing the linear characters for the group G.

Construct the table of irreducible characters for the group G. The characters are found using the Dixon-Schneider algorithm.

Given a group G and a subgroup H of G, construct the ordinary character afforded by the representation of G given by its action on the coset space of the subgroup H.

The natural R[G]-module for the matrix group G.

GModule(G, A) : GrpMat, AlgMat -> ModGrp

Let A be a matrix ring defined over the ring R and let G be a finite group defined on m generators. Let M denote the underlying module of A. Suppose there is a one-to-one correspondence between the generators of G and the generators [ A_1, ..., A_m ] of A. The function GModule creates the R[G]-module corresponding to an action of G on M defined by A, where the action of the i-th generator of G on M is given by A_i.

GModule(G, Q) : GrpMat, [ AlgMatElt ] -> ModGrp

Let A be a matrix ring defined over the ring R and let G be a finite group defined on m generators. Let M denote the underlying module of A. Given a sequence Q of m elements of A, the function GModule creates the R[G]-module corresponding to an action of G on M defined by Q, where the action of the i-th generator of G on M is given by Q[i].

GModule(G, A, B) : GrpMat, GrpMat, GrpMat -> ModGrp, Map

Let A and B be normal subgroups of G such that B is contained in A. Further, assume that A/B is elementary abelian of order p^n, p a prime. Let K denote the field of p elements. This function constructs a K[G]-module corresponding to the action of the group G on the elementary abelian section A/B of G. The map from A to the K[G]-module's underlying vector space is also returned.

The permutation module for G over the ring R defined by its action on the cosets of the subgroup H.