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 \(mu: A \rightarrow B\), returns the induced homomorphism \(A/\mathop{\hbox{\rm Rad}}^2(A) \rightarrow B/\mathop{\hbox{\rm Rad}}^2(B)\), where \(\mathop{\hbox{\rm Rad}}^2\) 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 \(\phi\). That is phi is the matrix of the induced homomorphism of \(X/Rad^2(X)\) to \(Y/Rad^2(Y).\)

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: Graded Properties of a Schur Algebra (ex-804d0a)

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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.

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Example: Reordering Primitive Idempotents (ex-27c999)

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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.

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