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.
The coefficient ring (base ring) of A.
The functions in this section can be applied to group algebras and their subalgebras of type AlgGrpSub.
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.
The group G for the group algebra (subalgebra) S.
The group algebra of which S is a subalgebra.
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.
The coefficient ring (base ring) of A.
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.
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.
Returns true if S is a left ideal of its group algebra; otherwise false.
Returns true if S is a right ideal of its group algebra; otherwise false.
Returns true if S is a (two-sided) ideal of its group algebra; otherwise false.
The centralizer of the subalgebra S of a group algebra A (in A).
The largest subalgebra T of A such that S is an ideal in T.
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.
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.
> 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