[GAP Forum] tensor product of representations

[Victor D. Mazurov]

Dear Alexander,

Thank you for your comprehensive help. Best wishes, Victor

2016-12-03 23:07 GMT+07:00 Hulpke,Alexander <Alexander.Hulpke at colostate.edu>
:

> 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,
> Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
> email: hulpke at colostate.edu, Phone: ++1-970-4914288
> http://www.math.colostate.edu/~hulpke
>
>
>


-- 
Victor Danilovich Mazurov
Institute of Mathematics
Novosibirsk 630090
Russia