Accessing Module Information

This section deals with the underlying vector space of a module M, which is a module over the algebra A.

Subsections

• The Underlying Vector Space
• The Algebra

The Underlying Vector Space

VectorSpace(M) : ModTupRng -> ModTupRng

Given an A-module M, return the underlying vector space of M.

M . i : ModTupRng, RngIntElt -> ModElt

Given an A-module M and a positive integer i, return the i-th generator of M.

CoefficientRing(M) : ModTupRng -> Rng

BaseRing(M) : ModTupRng -> Rng

Given an A-module M, where A is a an algebra over the field K, return K.

Generators(M) : ModTupRng -> { ModTupElt }

The generators for the A-module M, returned as a set.

Parent(u) : ModTupElt -> ModRng

Given an element u belonging to the A-module M, return M.

The Algebra

Algebra(M) : ModTupRng -> Rng

Given an A-module M, return the algebra A.

Action(M) : ModTupRng -> AlgMat

RightAction(M) : ModTupRng -> AlgMat

Given an A-module M, return the matrix algebra A giving the action of A on M.

MatrixGroup(M) : ModGrp -> GrpMat

Given an R[G]-module M, return the matrix group whose generators are the (invertible) generators of the acting algebra of M.

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

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

The i-th generator of the (right) acting matrix algebra for the module M.

NumberOfActionGenerators(M) : ModTupRng -> RngIntElt

Nagens(M) : ModTupRng -> RngIntElt

The number of action generators (the number of generators of the algebra) for the A-module M.

Group(M) : ModGrp -> Grp

Given an R[G]-module M, return the group G.

Example ModAlg_Access (H77E2)

> F2 := GF(2);
> F := MatrixAlgebra(F2, 6);
> A := sub< F |
>   [ 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 ] >;
> T := RModule(F2, 6);
> M := RModule(T, A);
> Dimension(M);
6
> BaseRing(M);
Finite field of size 2

We set R to be the name of the matrix ring associated with M. Using the generator subscript notation, we can access the matrices giving the (right) action of A.

> R := RightAction(M);
> R.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]
> R.2;
[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]

We display full details of the module.

> M: Maximal;
Module M of dimension 6 with base ring GF(2)
Generators of acting algebra:


[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]