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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.
For a group algebra A given in vector representation, construct the
associative structure constant algebra B isomorphic to A together
with the isomorphism A -> B.
The augmentation map of A.
That is, the map A -> R :
∑g ∈G rg * g -> ∑g ∈G rg.
The augmentation ideal of the group algebra A given in vector representation.
This is defined as the kernel of the augmentation map.
Given a group algebra A, return either "Vector" or "Terms"
depending on which representation is used for the elements of A.
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.
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.
BaseRing(A) : AlgGrp -> Rng
The coefficient ring (base ring) of A.
The functions in this section can be applied to group algebras and their
subalgebras of type AlgGrpSub.
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) : AlgGrp -> Grp
The group G for the group algebra (subalgebra) S.
GroupAlgebra(S) : AlgGrp -> AlgGrp
The group algebra of which S is a subalgebra.
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.
BaseRing(A) : AlgGrpSub -> Rng
The coefficient ring (base ring) of A.
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 x |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) : 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) : AlgGrp -> BoolElt
Returns true if S is a left ideal of its group algebra; otherwise false.
IsRightIdeal(S) : AlgGrp -> BoolElt
Returns true if S is a right ideal of its group algebra; otherwise false.
IsIdeal(S) : AlgGrp -> BoolElt
Returns true if S is a (two-sided) ideal of its group algebra; otherwise false.
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).
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) : 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 ∈S.
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 ∈S.
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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