Operations on Group Algebras and their Subalgebras

Operations on Group Algebras

The operations in this section can only be applied to a full group algebra. Functions accepting also a subalgebra of type AlgGrpSub are dealt with in the next section.

Algebra(A): AlgGrp -> AlgAss, Map: For a group algebra \(A\) given in vector representation, construct the associative structure constant algebra \(B\) isomorphic to \(A\) together with the isomorphism \(A \rightarrow B\).

AugmentationMap(A): AlgGrp -> Map: The augmentation map of \(A\). That is, the map \(A \rightarrow R : \sum_{g \in G} r_g * g \rightarrow \sum_{g \in G} r_g\).

AugmentationIdeal(A): AlgGrp -> AlgGrpSub: The augmentation ideal of the group algebra \(A\) given in vector representation. This is defined as the kernel of the augmentation map.

RepresentationType(A): AlgGrp -> MonStgElt: Given a group algebra \(A\), return either "Vector" or "Terms" depending on which representation is used for the elements of \(A\).

ChangeRepresentationType(A, Rep): AlgGrp, MonStgElt -> AlgGrp, Map: Given a group algebra \(A\), construct an isomorphic group algebra \(B\) in which the elements are represented as specified by Rep which may be "Vector" or "Terms", together with the homomorphism from \(A\) to \(B\).

ConstructTable(A): AlgGrp: Procedure which, given a group algebra \(A = R[G]\) in vector representation, constructs the multiplication table for the group \(G\) to speed up multiplication in \(A\). If the multiplication table already exists, nothing is done.

CoefficientRing(A): AlgGrp -> Rng
BaseRing(A): AlgGrp -> Rng: The coefficient ring (base ring) of A.

Operations on Subalgebras of Group Algebras

The functions in this section can be applied to group algebras and their subalgebras of type AlgGrpSub.

S ! 1: AlgGrpSub -> Grp
S ! 1: AlgGrp -> Grp: Create the identity element of the group algebra (subalgebra) \(S\). Note that for a proper subalgebra of the full group algebra this may be different from the identity element of the group.

Group(S): AlgGrpSub -> Grp
Group(S): AlgGrp -> Grp: The group \(G\) for the group algebra (subalgebra) \(S\).

GroupAlgebra(S): AlgGrpSub -> AlgGrp
GroupAlgebra(S): AlgGrp -> AlgGrp: The group algebra of which \(S\) is a subalgebra.

Module(S): AlgGrpSub -> ModTupRng, Map
Module(S): AlgGrp -> ModTupRng, Map: For a subalgebra \(S\) of the group algebra \(A = R[G]\), return the submodule of the module underlying \(A\) which corresponds to \(S\). This is an \(R\)-module of dimension Dimension(S) and degree Dimension(A). Also returns (as a second return value) the natural map from the subalgebra to the module.

CoefficientRing(A): AlgGrpSub -> Rng
BaseRing(A): AlgGrpSub -> Rng: The coefficient ring (base ring) of A.

BasisMatrix(S): AlgGrpSub -> ModMatRngElt
BasisMatrix(S): AlgGrp -> ModMatRngElt: For a subalgebra \(S\) of the group algebra \(A = R[G]\) return the coefficient matrix of the basis of \(S\) with respect to the basis of \(A\). If \(S\) has dimension \(m\) this is an \(m \times |G|\)-matrix over \(R\) where the \(i\)-th row are the coefficients of the \(i\)-th basis vector of \(S\) with respect to the fixed basis of \(A\).

Coordinates(S, a): AlgGrpSub, AlgGrpElt -> [ RingElt ]
Coordinates(S, a): AlgGrp, AlgGrpElt -> [ RingElt ]: Given an element \(a\) which lies in the subalgebra \(S\), return a sequence giving the coordinates of \(a\) with respect to the basis of \(S\).

IsLeftIdeal(S): AlgGrpSub -> BoolElt
IsLeftIdeal(S): AlgGrp -> BoolElt: Returns true if \(S\) is a left ideal of its group algebra; otherwise false.

IsRightIdeal(S): AlgGrpSub -> BoolElt
IsRightIdeal(S): AlgGrp -> BoolElt: Returns true if \(S\) is a right ideal of its group algebra; otherwise false.

IsIdeal(S): AlgGrpSub -> BoolElt
IsIdeal(S): AlgGrp -> BoolElt: Returns true if \(S\) is a (two-sided) ideal of its group algebra; otherwise false.

Centraliser(S): AlgGrpSub -> AlgGrpSub
Centralizer(S): AlgGrpSub -> AlgGrpSub
Centraliser(S): AlgGrp -> AlgGrpSub
Centralizer(S): AlgGrp -> AlgGrpSub: The centralizer of the subalgebra \(S\) of a group algebra \(A\) (in \(A\)).

Idealiser(S): AlgGrpSub -> AlgGrpSub
Idealizer(S): AlgGrpSub -> AlgGrpSub
Idealiser(S): AlgGrp -> AlgGrpSub
Idealizer(S): AlgGrp -> AlgGrpSub: The largest subalgebra \(T\) of \(A\) such that \(S\) is an ideal in \(T\).

LeftAnnihilator(S): AlgGrpSub -> AlgGrpSub
LeftAnnihilator(S): AlgGrp -> AlgGrpSub: For a subalgebra \(S\) of the group algebra \(A\) construct the left annihilator of \(S\), that is, the subalgebra of \(A\) consisting of all elements \(a\) such that \(a*s = 0\) for all \(s \in S\).

RightAnnihilator(S): AlgGrpSub -> AlgGrpSub
RightAnnihilator(S): AlgGrp -> AlgGrpSub: For a subalgebra \(S\) of the group algebra \(A\) construct the right annihilator of \(S\), that is, the subalgebra of \(A\) consisting of all elements \(a\) such that \(s*a = 0\) for all \(s \in S\).

Example: Jacobson Radical of a Group Algebra (ex-71b601)

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We construct the group algebra of an elementary abelian group over \(GF(2)\) and get its Jacobson ideal.

> A := AbelianGroup([2,2,2,2,2]);
> FG := GroupAlgebra(GF(2), A);
> J := JacobsonRadical(FG);
> J;
Ideal of dimension 31 of the group algebra FG

We now check that the Jacobson radical is nilpotent and get its nilpotency class.

> JPow := [ J ];
> I := J;
> while Dimension(I) ne 0 do
>     I := I*J;
>     Append(~JPow, I);
> end while;
> [ Dimension(I) : I in JPow ];
[ 31, 26, 16, 6, 1, 0 ]

Thus, \(J\) is nilpotent of class 6. However, every non-zero element of \(J\) is of course nilpotent of class 2.

> IsNilpotent(Random(J));
true 2

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Operations on Group Algebras and their Subalgebras#