# 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 Homomorphism (ex-804d0a)`

In this examples we investigate properties of the basic algebra of a Schur algebra.

```magma
> 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: GradedHomomorphisms 2 (ex-27c999)`

Here we see how to rearrange an algebra by changing the ordering on the primitive idempotents.

```magma
> 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.