Dear Forum, Dear Victor Mazurov,
> On Dec 2, 2016, at 10:24 PM, Victor D. Mazurov <mazu...@math.nsc.ru> wrote:
>
> Dear forum,
>
> How can I get a homomorphism from given representation of finite group to
> the another one?
>
> Example: By Atlas of FGR,
> Matrices
> […]
> generate a 4-dimensional representation U of alternating group A_8 over a
> field of order 2 and
> matrices
>
> […]
>
> generate a 6-dimensional representation V of A_8 over a field of order
> 2.
>
If you get matrices from the online ATLAS, you are in luck in that they are
always given on the same generators, that is isomorphisms will simply map the
one generating set to the other. For example, you could use
hom:=GroupHomomorphismByImages(U,V,GeneratorsOfGroup(U),GeneratorsOfGroup(V));
to construct such an isomorphism. You can apply it with `Image(how,elm)` on
elements or subgroups.
> How can I calculate H=Hom(U\otimes U,V) and, if H\ne 0, a homomorphism of
> U\otimes U onto V?
Do you mean by U\otimes U the tensor-square representation? If so, you do the
same (with generators still fitting)
gap> tens:=List(GeneratorsOfGroup(U),
> x->KroneckerProduct(x,x));
gap> A:=Group(tens);
gap> hom:=GroupHomomorphismByImages(A,V,GeneratorsOfGroup(A),GeneratorsOfGroup($
If the generators do not agree, you would have to do an explicit homomorphism
search. E.g. (forcing different generators:
gap> B:=Group(Random(U),Random(U));Size(B);
<matrix group with 2 generators>
20160
gap> IsomorphismGroups(B,V);
CompositionMapping(
[ (2,9)(4,11)(6,13)(8,15), (2,7,6,10,12)(3,11,8,4,13)(5,16,15,9,14) ] ->
[ <an immutable 6x6 matrix over GF2>, <an immutable 6x6 matrix over GF2> ],
<action isomorphism> )
All the best,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hul...@colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
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