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Homomorphisms of Modules


Homomorphisms of Modules








Let M and N be R[G]-modules.  Then Hom_(R[G])(M, N) consists of
all R-homomorphisms from M to N which preserve the action of G.
The type of such (matrix) homomorphisms, called G-homs, is
ModMatGrpElt.



EndomorphismAlgebra(M) : ModRng -> AlgMat



Given a R-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_(R)(M). 



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



Given an m-dimensional K[G]-module M and an n-dimensional 
K[G]-module N, K a finite field, construct Hom_(K[G])(M, N) as
a submodule of Hom_K(M, N).



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



Given an m-dimensional K[G]-module M and an n-dimensional 
K[G]-module N, K a finite field, construct Hom_(L[G])(M, N) as
a submodule of Hom_K(M, N) where L is the centralizing field of
M.



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



Given an m-dimensional R-module M and an n-dimensional 
R-module N, construct Hom_R(M, N). 



IsGHom(X) : ModMatElt -> BoolElt



Given matrix X from Hom_K(M, N), where M and N are G-modules,
return whether X is a G-hom, i.e. X is in GHom(M, N).



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 Hom_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.



Morphism(M, N) : ModRng, ModRng -> ModMatRngElt



(a) If the R-module M was created as a submodule of the module N,
return the inclusion homomorphism phi : M -> N as an element of
Hom_R(M, N). Thus phi gives the correspondence between elements of
M (represented with respect to the standard basis of M) and elements
for N.


(b) If the R-module N was created as a quotient module of the module M,
return the natural homomorphism phi : M -> N as an element of
Hom_R(M, N). Thus phi gives the correspondence between elements of M
and elements of N (represented with respect to the standard basis for N).



ImageWithBasis(X, M) : ModMatRngElt, ModRng -> ModRng



Given a matrix X which describes a basis for a submodule of M (i.e. X
describes an invariant subspace under the action of M), return the 
submodule S of M such that the morphism of S into M is X.





Example RMod_EndoRing (H42E20)

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) >;
> K := GaloisField(3);
> P := PermutationModule(G, K);
> ER := EndomorphismAlgebra(P);
> ER : Maximal;
Matrix Algebra of degree 12 and dimension 3 over GF(3)
Basis:




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




[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]




[0 0 0 1 1 1 1 1 1 1 1 1]
[0 0 0 1 1 1 1 1 1 1 1 1]
[0 0 0 1 1 1 1 1 1 1 1 1]
[1 1 1 0 0 0 1 1 1 1 1 1]
[1 1 1 0 0 0 1 1 1 1 1 1]
[1 1 1 0 0 0 1 1 1 1 1 1]
[1 1 1 1 1 1 0 0 0 1 1 1]
[1 1 1 1 1 1 0 0 0 1 1 1]
[1 1 1 1 1 1 0 0 0 1 1 1]
[1 1 1 1 1 1 1 1 1 0 0 0]
[1 1 1 1 1 1 1 1 1 0 0 0]
[1 1 1 1 1 1 1 1 1 0 0 0]
> // Thus, the permutation module P has 27 endomorphisms.
> // We now find a maximal submodule M of P and calculate GHom(M, P).
> MS := MaximalSubmodules(P);
> MS;
[
    GModule of dimension 9 with base ring GF(3),
    GModule of dimension 11 with base ring GF(3)
]
> M := MS[1];
> HR := GHom(P, M);
> HR: 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 := (K!2)*HR.1 + HR.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 will 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]





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