The Representation Afforded by a K[G]-module

GModuleAction(M) : ModGrp -> Map(Hom)

Given a K[G]-module M, return the action of G on M as homomorphism f of G into the matrix group GL_n(K).

Representation(M) : ModGrp -> Map(Hom)

Given a K[G]-module M, return the action of G on M as homomorphism f of G into the matrix algebra M_n(K).

ActionGenerator(M, i) : ModGrp, RngIntElt -> AlgMatElt

RightActionGenerator(M, i) : ModGrp, RngIntElt -> AlgMatElt

The i-th generator of the (right) acting matrix algebra for the module M. That is, the image of the i-th group generator in the corresponding representation.

ActionGenerators(M) : ModGrp -> [ AlgMatElt ]

Return the matrices giving the action on M as a sequence. These are the images of the generators of the group in the corresponding representation.

NumberOfActionGenerators(M) : ModGrp -> RngIntElt

Nagens(M) : ModGrp -> RngIntElt

The number of action generators (the number of generators of the algebra) for the R[G]-module M.

ActionGroup(M) : ModGrp -> GrpMat

The matrix group generated by the action generators of M.

Example ModGrp_Representation (H73E8)

> A7 := AlternatingGroup(7);
> M  := PermutationModule(A7, Vector(GF(11), [1,0,1,0,1,0,1]));
> phi := Representation(M);
> [ Trace(phi(c[3])) : c in Classes(A7) ]; 
[ 7, 3, 4, 1, 1, 2, 0, 0, 0 ]

Example ModGrp_Dual (H73E9)

> DualModule := function(M)
>       G := Group(M);
>       f := Representation(M);
>       return GModule(G, [ Transpose(f(G.i))^-1 : i in [1 .. Ngens(G)] ]);
> end function;