Minimal Forms and Gradings

There are some standard ways of rewriting an algebra to an isomorphic form. We can also constructed the associated graded algebra of a basic algebra. Also included here is a function for rearranging the orders of idempotents in a basic algebra.

The first intrinsic is used extensively in the programs for computing automorphisms and isomorphisms.

MinimalGeneratorForm(A) : AlgBas -> Rec
Returns a record consisting of an isomorphic basic algebra having the property that it is generated by a minimal number of elements and the projective modules are filtered by radical layers. The record consists of the following fields:
(a)
The algebra in minimal form (field algebra).
(b)
The map from the minimal generator form algebra to A (field Homomorphism).
(c)
The inverse map from A to the minimal generator form algebra (field InverseHomomorphism).
(d)
The dimensions of the radical layer (field RadicalDimensions).
(e)
The dimensions of the filtration of the top radical layer by the socle layer (field FilterDimensions).
MinimalGeneratorFormAlgebra(A) : AlgBas -> AlgBas
Returns an isomorphic algebra having minimal generator form.
AssociatedGradedAlgebra(A) : AlgBas -> AlgBas
Returns the basic algebra that is isomorphic to the associated graded algebra of A.
GradedCapHomomorphism(A) : AlgBas -> ModMatFldElt
Returns the matrix of the map from A/Rad(A) to X/Rad(X) where X is the associated graded algebra of A.
GradedCapHomomorphism(A, B, mu) : AlgBas, AlgBas, ModMatFldElt -> ModMatFldElt
Given an algebra homomorphism mu: A -> B, returns the induced homomorphism A/Rad2(A) -> B/Rad2(B), where Rad2 is the second power of the Jacobson radical.
BuildHomomorphismFromGradedCap(A, B, phi) : AlgBas, AlgBas, ModMatFldElt -> ModMatFldElt
Returns the graded homomorphism from the associated graded algebra X of A to the associated graded algebra Y of B, whose cap is φ. That is phi is the matrix of the induced homomorphism of X/Rad2(X) to Y/Rad2(Y).
ChangeIdempotents(A, S) : AlgBas, SeqEnum -> AlgBas, Map
ChangeIdempotents(A, S) : AlgBas, GrpPermElt -> AlgBas, Map
Returns the basic algebra isomorphic to A, obtained by permuting the order of the idempotents by the permutation S. The permutation S can be given as an element of the symmetric group or as the sequence of images of the permutation.

Example AlgBas_GradedHomomorphism (H92E7)

In this examples we investigate properties of the basic algebra of a Schur algebra. 
> A := BasicAlgebraOfSchurAlgebra(3,6,GF(3));
> A;
Basic algebra of dimension 48 over GF(3)
Number of projective modules: 7
Number of generators: 21
> B := BasicAlgebraOfExtAlgebra(A,10);
> B;
Basic algebra of dimension 98 over GF(3)
Number of projective modules: 7
Number of generators: 21
> C := BasicAlgebraOfExtAlgebra(B,10);
> C;
Basic algebra of dimension 48 over GF(3)
Number of projective modules: 7
Number of generators: 21
> boo,mat := IsIsomorphic(A,C);
> boo;
true
> IsAlgebraHomomorphism(mat);
true
So we see that A is isomorphic to its double ext-algebra, and it is graded since the double ext-algebra is graded. Thus we know that the basic algebra A is a Koszul algebra.

Example AlgBas_GradedHomomorphisms-2 (H92E8)

Here we see how to rearrange an algebra by changing the ordering on the primitive idempotents. 
> A := BasicAlgebraOfSchurAlgebra(3,5,GF(3));
> A;
Basic algebra of dimension 11 over GF(3)
Number of projective modules: 5
Number of generators: 9
> B, uu  := ChangeIdempotents(A,[2,4,5,1,3]);
> B;
Basic algebra of dimension 11 over GF(3)
Number of projective modules: 5
Number of generators: 9
> DimensionsOfProjectiveModules(A);
[ 2, 3, 2, 1, 3 ]
> DimensionsOfProjectiveModules(B);
[ 3, 1, 3, 2, 2 ]
> IsAlgebraHomomorphism(A,B,uu);
true
> uu;
[0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0]
[1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0]
We see that uu is a permutation matrix.
V2.28, 13 July 2023