Accessing Module Information#
This section deals with the underlying vector space of a module \(M\), which is a module over the algebra \(A\).
The Underlying Vector Space#
- M . i: ModRng, RngIntElt -> ModElt#
Given an \(A\)-module \(M\) and a positive integer \(i\), return the \(i\)-th generator of \(M\).
- CoefficientRing(M): ModRng -> Rng#
- BaseRing(M): ModRng -> Rng#
Given an \(A\)-module \(M\), where \(A\) is an algebra over the field \(K\), return \(K\).
- Generators(M): ModRng -> { ModRngElt }#
The generators for the \(A\)-module \(M\), returned as a set.
- Parent(u): ModRngElt -> ModRng#
Given an element \(u\) belonging to the \(A\)-module \(M\), return \(M\).
The Action Algebra#
- Action(M): ModRng -> AlgMat#
- RightAction(M): ModRng -> AlgMat#
Given an \(A\)-module \(M\), return the matrix algebra \(A\) giving the action of \(A\) on \(M\).
- MatrixGroup(M): ModGrp -> GrpMat#
Check: BoolElt Default: true
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): ModRng, RngIntElt -> AlgMatElt#
The \(i\)-th generator of the (right) acting matrix algebra for the module \(M\).
- NumberOfActionGenerators(M): ModRng -> RngIntElt#
- Ngens(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: Access (ex-891476)#
We illustrate the use of several of these access functions by applying them to the \(6\)-dimensional representation of a matrix algebra defined over \({\bf F}_{2}\).
> 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]