A set of functions are provided for computing with the natural G-module M of a matrix group G. Many of these functions are similar to those in the general module chapter.

GModule(G) : GrpMat -> ModGrp

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

Given a matrix group G, return true iff G acts irreducibly on its natural module M. If G acts reducibly on M, a proper submodule S of M is also returned.

Given a matrix group G and a submodule S of the natural module M of G, return the action homomorphism f of G on S, together with the image of f.

Given a matrix group G and a submodule S of the natural module M of G, return the image of the action homomorphism of G on S.

Given a matrix group G and a submodule S of the natural module M of G, return the quotient action homomorphism f of G on S, together with the image of f.

Given a matrix group G and a submodule S of the natural module M of G, return the quotient image of the action homomorphism of G on S.

Given a matrix group G, return true iff G acts absolutely irreducibly on its natural module M. Return also the (matrix algebra) generator of the endomorphism algebra E of M (which is always a field), as well as the dimension of E

Given an irreducible matrix group G, construct the isomorphic reduced-degree absolute representation of G, which is over the absolute field of the natural module M of G and is absolutely irreducible.

Given a matrix group G over a finite field, return the minimal field to which G can be restricted (a subfield of the coefficient field of G).