Dear Alexander, Thank you for your comprehensive help. Best wishes, Victor
2016-12-03 23:07 GMT+07:00 Hulpke,Alexander <alexander.hul...@colostate.edu> : > 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 > > > -- Victor Danilovich Mazurov Institute of Mathematics Novosibirsk 630090 Russia _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
