[GAP Forum] tensor product of representations

[Hulpke,Alexander]

Dear Forum, Dear Victor Mazurov,


> On Dec 2, 2016, at 10:24 PM, Victor D. Mazurov <mazurov at 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,
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email: hulpke at colostate.edu, Phone: ++1-970-4914288
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