[GAP Forum] Double cosets of orthogonal groups (rather than permutation groups)

[Alexander Hulpke]

Dear  Forum,

It's double coset week!

On Jan 12, 2011, at 1/12/11 1:07, Katie Morrison wrote:

> I am trying to compute double cosets within the orthogonal group with
> similitudes (which I am taking to be the set of matrices that satisfy
> MM^T=\lambda *I where I is the identity and \lambda is a non-zero scalar).
> I have obtained the orthogonal group with similitudes by starting with
> GO(n,q), conjugating it by an appropriate change of basis matrix to get the
> matrices that preserve the standard dot product rather than the bilinear
> form that GAP uses, and then taking the normalizer of this group in
> GL(n,q).
> Using this construction of the orthogonal group with similitudes,
> I would like to then find double cosets of certain subgroups of this, but as
> far as I know GAP only has the double-cosets function implemented for
> permutation groups.

Since the normalizer calculation succeeded, it seems your parameters are small enough to allow for the construction of a permutation representation. In this situation you will be able to use this representation and permutation group functionality to get the results. For example, if G is the normalizer and A and B are the two subgroups, the following commands would get you double coset information in the matrix group:

hom:=IsomorphismPermGroup(G);
Gperm:=Image(hom,G);
Aperm:=Image(hom,A);
Bperm:=Image(hom,B);
dc:=DoubleCosetRepsAndSizes(Gperm,Aperm,Bperm);
List(dc,x->[PreImagesRepresentative(hom,x[1]),x[2]]);

Best wishes,

   Alexander Hulpke



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