[GAP Forum] Re: Computing image of a homomorphism

[Ravi Kulkarni]

Dear J"urgen,
  I mailed a query to the GAP Forum a few days ago, but have not
received any reply. Is it possible for you to look at this?
  Ravi Kulkarni

On Sat, Oct 31, 2009 at 6:56 PM, Ravi Kulkarni <ravi.kulk at gmail.com> wrote:
> Dear GAP Forum,
>  I need to calculate the image of a homomorphism of a matrix group:
>
> gap> m1 := [ [0,1,0],[-1,0,0],[0,0,1] ];;
> gap> m2 := [ [0,1,0],[0,0,1],[1,0,0] ];;
> gap> O := Group(m1,m2);; # one of the two 3-dim irrep of SymmetricGroup(4) (S_4)
>
> # compute the five conjugacy classes and combine them into one list
> # skip some code...
> gap> ccall := Concatenation(cc1,cc2,cc3,cc4,cc5);
>
> I want to calculate the matrices corresponding to the two dimensional
> irrep of S_4. Call this W. Examining the character table (and some
> computation) shows that W is a representation of the quotient of S_4
> by the group generated by the elements of the fifth conjugacy class
> (which has character 2 in the W representation).
>
> gap> H := Group(cc5);;
> gap> IsNormal(O,H);
> true
>
> gap> OmodH := FactorGroup(O,H);
> Group([ f1, f2^2 ])
>
> gap> hom := NaturalHomomorphismByNormalSubgroup(O,H);
> CompositionMapping( [ (1,2,6,5), (1,2,3)(4,6,5) ] ->
> [ f1, f2^2 ], <action isomorphism> )
>
> Compute the images of elements of O:
>
> gap> imghom := List([1..Size(O)], i -> Image(hom,ccall[i]));
> [ <identity> of ..., f1*f2, f1*f2, f1*f2^2, f1*f2^2, f1, f1, f2^2, f2, f2,
>  f2^2, f2, f2^2, f2, f2^2, f1, f1, f1*f2, f1*f2, f1*f2^2, f1*f2^2,
>  <identity> of ..., <identity> of ..., <identity> of ... ]
>
> Now, what I would like to have are the 2x2 matrices corresponding to
> each of the elements of "imghom". I can guess what f1 and f2 are in
> this example, but would be wary of doing so for a larger group.
>
> Is there a systematic way of getting the matrices f1 and f2?
>
> Thanks,
>  Ravi
>