Construction of an A-Module

Subsections

• General Constructions
• Constructions for K[G]-Modules

General Constructions

RModule(A) : AlgMat -> ModTupRng

Given a subalgebra A of M_n(K), create the right A-module M with underlying vector space K^((n)), where the action of a in A is given by m * a, m in M.

RModule(Q) : [ AlgMatElt ] -> ModTupRng

Given the subalgebra A of M_n(K) generated by the terms of the sequence Q, create the right A-module M with underlying vector space K^((n)), where the action of a in A is given by m * a, m in M.

Example ModAlg_CreateK6 (H77E1)

     [0 1 1 0 1 0]
     [0 0 1 1 1 1]
     [1 0 0 1 0 1]
     [0 0 0 1 0 0]
     [0 0 0 0 1 0]
     [0 0 0 0 0 1],

     [1 0 0 1 0 1]
     [0 1 0 0 1 1]
     [0 1 1 1 1 0]
     [0 0 0 1 1 0]
     [0 0 0 1 0 1]
     [0 1 0 1 0 0]

> A := MatrixAlgebra<GF(2), 6 |
>   [ 1,0,0,1,0,1, 
>     0,1,0,0,1,1, 
>     0,1,1,1,1,0, 
>     0,0,0,1,1,0, 
>     0,0,0,1,0,1,
>     0,1,0,1,0,0 ],
>   [ 0,1,1,0,1,0,
>     0,0,1,1,1,1,
>     1,0,0,1,0,1,
>     0,0,0,1,0,0,
>     0,0,0,0,1,0,
>     0,0,0,0,0,1 ] >;
> M := RModule(A);
> M;
RModule M of dimension 6 over GF(2)

Constructions for K[G]-Modules

Although K[G]-modules are discussed in the next chapter, it is convenient to use them as examples in this chapter and so we give two basic constructions here. The reader is referred to the K[G]-module chapter for many other techniques for constructing these modules.

GModule(G, Q) : Grp, [ GrpMatElt ] -> ModGrp

Let G be a group defined on r generators and let Q be a sequence of r invertible elements of M_n(K) or GL(n, K). It is assumed that the mapping from G to Q defined by phi(G.i) |-> Q[i], for i = 1, ..., r, is a group homomorphism from G into GL(n, K). The function constructs a K[G]-module M of dimension n, where the action of the generators of G is given by the terms of Q.

PermutationModule(G, K) : Grp, Fld -> ModGrp

Given a permutation group G and a field K, create the natural permutation module for G over K.