Homomorphisms

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Homomorphisms

Let M and N be A-modules where A is an algebra defined over a field K. Then öm_(A)(M, N) consists of all K-homomorphisms from M to N which commute with the action of A. The type of such (matrix) homomorphisms, called A-homs, is ModMatGrpElt.

The functions is this section currently apply only in the case in which A is an algebra over a finite field.

Subsections

Creating Homomorphisms

hom< M -> N | X > : ModRng, ModRng, ModMatElt -> ModMatRng

Given R-modules M and N, create the (map) homomorphism from M to N given by matrix X.

H ! f : ModMatRng, Map -> ModMatRngElt

Given matrix space H, which is öm_R(M, N) for R-modules M and N, together with a homomorphism f from M to N, create the matrix corresponding to the map f.

IsModuleHomomorphism(X) : ModMatElt -> BoolElt

Given a matrix X belonging to öm_K(M, N), where M and N are A-modules, return true if X is an A-homomorphism.

öm(M, N)

Hom(M, N) : ModRng, ModRng -> ModMatRng

Given A-modules M and N, construct the vector space of homomorphisms, öm_K(M, N), where K is the field over which A is defined.

AHom(M, N) : ModGrp, ModGrp -> ModMatGrp

Given A-modules M and N, construct the vector space of homomorphisms, öm_A(M, N), as a submodule of öm_K(M, N).

GHomOverCentralizingField(M, N) : ModGrp, ModGrp -> ModMatGrp

Given A-modules M and N, construct öm_(L[G])(M, N) as a subspace of öm_K(M, N) where L is the centralizing field of M.

Example `ModAlg_EndoRing (H76E21)`

We construct a 12-dimensional module M and a 9-dimensional submodule P of M for a soluble group of order 648 over GF(3). We then construct H = öm_A(M, N) and perform map operations on elements of H.

> G := PermutationGroup< 12 |
>         (1,6,7)(2,5,8,3,4,9)(11,12),
>         (1,3)(4,9,12)(5,8,10,6,7,11) >;
> K := GF(3);
> P := PermutationModule(G, K);
> M := sub< P | [1,0,0,0,0,1,0,0,1,0,0,1] >;
> M;
GModule M of dimension 9 over GF(3)
> H := AHom(P, M);
> H: Maximal;
KMatrixSpace of 12 by 9 GHom matrices and dimension 2 over GF(3)
Echelonized basis:
[1 1 1 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0]
[0 0 0 1 1 0 0 0 0]
[0 0 0 1 1 0 0 0 0]
[0 0 0 1 1 0 0 0 0]
[0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 1 1]


[0 0 0 1 1 1 1 1 1]
[0 0 0 1 1 1 1 1 1]
[0 0 0 1 1 1 1 1 1]
[1 1 1 0 0 1 1 1 1]
[1 1 1 0 0 1 1 1 1]
[1 1 1 0 0 1 1 1 1]
[1 1 1 1 1 0 0 1 1]
[1 1 1 1 1 0 0 1 1]
[1 1 1 1 1 0 0 1 1]
[1 1 1 1 1 1 1 0 0]
[1 1 1 1 1 1 1 0 0]
[1 1 1 1 1 1 1 0 0]
> // We write down a random homomorphism from M to P.
> f := 2*H.1 + H.2;
> f;
[2 2 2 1 1 1 1 1 1]
[2 2 2 1 1 1 1 1 1]
[2 2 2 1 1 1 1 1 1]
[1 1 1 2 2 1 1 1 1]
[1 1 1 2 2 1 1 1 1]
[1 1 1 2 2 1 1 1 1]
[1 1 1 1 1 2 2 1 1]
[1 1 1 1 1 2 2 1 1]
[1 1 1 1 1 2 2 1 1]
[1 1 1 1 1 1 1 2 2]
[1 1 1 1 1 1 1 2 2]
[1 1 1 1 1 1 1 2 2]
> Ker := Kernel(f);
> Ker;
GModule Ker of dimension 8 with base ring GF(3)

If we print the morphism associated with Ker, we see generators for Ker as a submodule of P.

> Morphism(Ker, P);
[1 0 2 0 0 0 0 0 0 0 0 0]
[0 1 2 0 0 0 0 0 0 0 0 0]
[0 0 0 1 0 2 0 0 0 0 0 0]
[0 0 0 0 1 2 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 2 0 0 0]
[0 0 0 0 0 0 0 1 2 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 2]
[0 0 0 0 0 0 0 0 0 0 1 2]
> // Examine the image of f and its morphism to P.
> Im := Image(f);
> Im;
GModule Im of dimension 4 with base ring GF(3)
> Morphism(Im, P);
[1 1 1 0 0 0 0 0 0 0 0 0]
[0 0 0 1 1 1 0 0 0 0 0 0]
[0 0 0 0 0 0 1 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1 1 1]

Example `ModAlg_CreateHomGHom (H76E22)`

We construct a G-homomorphism module H_1 for a G-module and then the homomorphism module H = öm(H_1, H_1) with right matrix action which is equivalent to (the right representation of) the endomorphism module of H.

> P := GModule(CyclicGroup(11), GF(3));
> F := Constituents(P);
> F;
[
    GModule of dimension 1 over GF(3),
    GModule of dimension 5 over GF(3),
    GModule of dimension 5 over GF(3)
]
> H1 := GHom(P, F[2]);
> H1;
KMatrixSpace of 2 by 3 matrices and dimension 1 over Rational Field
> H := Hom(H1, H1, "right");
> H: Maximal;
KMatrixSpace of 5 by 5 matrices and dimension 5 over GF(3)
Echelonized basis:


[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]


[0 1 0 0 0]
[1 1 1 2 1]
[2 0 2 1 1]
[2 1 0 0 0]
[0 2 1 0 0]


[0 0 1 0 0]
[2 0 2 1 1]
[2 2 2 2 2]
[2 0 1 0 2]
[1 0 1 2 1]


[0 0 0 1 0]
[2 1 0 0 0]
[2 0 1 0 2]
[2 2 1 2 2]
[2 1 1 0 1]


[0 0 0 0 1]
[0 2 1 0 0]
[1 0 1 2 1]
[2 1 1 0 1]
[2 1 0 0 2]

Endomorphisms and Automorphisms

EndomorphismAlgebra(M) : ModRng -> AlgMat

Given a A-module M with base ring K, construct End(M) as a subring E of the complete matrix ring K^((n x n)). The generators constructed for E form a K-basis for End_(A)(M).

AutomorphismGroup(M) : ModRng -> AlgMat

Given a A-module M with base ring K, construct Aut(M) as a subgroup G of the general linear group GL(n, K). Thus, G is the group of units of End(M).

IsIsomorphic(M, N) : ModRng, ModRng -> BoolElt, AlgMatElt

True if the R-modules M and N are isomorphic, false otherwise. If M and N are isomorphic, the function also returns a matrix T such that M^(T) = N.

Example `ModAlg_EndoRing (H76E23)`

We construct the endomorphism ring for a permutation module over GF(3) for a soluble group of order 648.

> G := PermutationGroup< 12 |
>         (1,6,7)(2,5,8,3,4,9)(11,12),
>         (1,3)(4,9,12)(5,8,10,6,7,11) >;
> P := PermutationModule(G, GF(3));
> time End := EndomorphismAlgebra(P);
Time: 0.100
> End;
Matrix Algebra of degree 12 and dimension 3 over GF(3)

Thus, the permutation module P has 27 endomorphisms.

> time Aut := AutomorphismGroup(P);
Time: 0.049
> Aut;
MatrixGroup(12, GF(3))
Generators:
    [1 0 0 1 1 1 1 1 1 1 1 1]
    [0 1 0 1 1 1 1 1 1 1 1 1]
    [0 0 1 1 1 1 1 1 1 1 1 1]
    [1 1 1 1 0 0 1 1 1 1 1 1]
    [1 1 1 0 1 0 1 1 1 1 1 1]
    [1 1 1 0 0 1 1 1 1 1 1 1]
    [1 1 1 1 1 1 1 0 0 1 1 1]
    [1 1 1 1 1 1 0 1 0 1 1 1]
    [1 1 1 1 1 1 0 0 1 1 1 1]
    [1 1 1 1 1 1 1 1 1 1 0 0]
    [1 1 1 1 1 1 1 1 1 0 1 0]
    [1 1 1 1 1 1 1 1 1 0 0 1]


    [2 1 1 0 0 0 0 0 0 0 0 0]
    [1 2 1 0 0 0 0 0 0 0 0 0]
    [1 1 2 0 0 0 0 0 0 0 0 0]
    [0 0 0 2 1 1 0 0 0 0 0 0]
    [0 0 0 1 2 1 0 0 0 0 0 0]
    [0 0 0 1 1 2 0 0 0 0 0 0]
    [0 0 0 0 0 0 2 1 1 0 0 0]
    [0 0 0 0 0 0 1 2 1 0 0 0]
    [0 0 0 0 0 0 1 1 2 0 0 0]
    [0 0 0 0 0 0 0 0 0 2 1 1]
    [0 0 0 0 0 0 0 0 0 1 2 1]
    [0 0 0 0 0 0 0 0 0 1 1 2]


    [0 1 1 0 0 0 0 0 0 0 0 0]
    [1 0 1 0 0 0 0 0 0 0 0 0]
    [1 1 0 0 0 0 0 0 0 0 0 0]
    [0 0 0 0 1 1 0 0 0 0 0 0]
    [0 0 0 1 0 1 0 0 0 0 0 0]
    [0 0 0 1 1 0 0 0 0 0 0 0]
    [0 0 0 0 0 0 0 1 1 0 0 0]
    [0 0 0 0 0 0 1 0 1 0 0 0]
    [0 0 0 0 0 0 1 1 0 0 0 0]
    [0 0 0 0 0 0 0 0 0 0 1 1]
    [0 0 0 0 0 0 0 0 0 1 0 1]
    [0 0 0 0 0 0 0 0 0 1 1 0]
> #Aut;
18
> IsAbelian(Aut);
true
> AbelianInvariants(Aut);
[ 2, 3, 3 ]

The module has 18 automorphisms. The automorphism group is isomorphic to the abelian group Z_2 x Z_3 x Z_3.

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