[GAP Forum] multiplication tables of double coset representatives

[Asst. Prof. Dmitrii (Dima) Pasechnik]

Dear Pooja Singla,

I would u talk about the equivalent setup of the algebra of orbitals
of G in its action on the (say, left) cosets of H  in G.
You can use GRAPE (a GAP package) to compute the
multiplication coefficients, or a (not yet official) GAP package
cohcfg:
http://www1.spms.ntu.edu.sg/~dima/software/cohcfg-a1.tgz
(you need to install it by, say, unpacking into the pkg/ subdirectory
of your GAP installation)

It has functions to compute the multiplication coefficients (a.k.a.
structure constants):

#(here is an example for S_5 acting on pairs of {1..5})
gap> LoadPackage("cohcfg");;
gap>g:=Action(SymmetricGroup(5),Orbit(SymmetricGroup(5),[1,2],OnSets),OnSets);;
gap> A:=TwoOrbitNumbers(g);;
# (here  (1,2,3)=(k,i,j))
gap> StructureConstantTwoOrbitsAlgebra(A,1,2,3);
0

# computing the regular representation of the algebra, i.e. an
#isomorphism to a certain algebra of nxn marices
gap> RegularRepresentationMatrices(A);
[ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
  [ [ 0, 6, 0 ], [ 1, 3, 2 ], [ 0, 4, 2 ] ],
  [ [ 0, 0, 3 ], [ 0, 2, 1 ], [ 1, 2, 0 ] ] ]


(sorry, there is still no manual, and the function names are going  to
be changed, but the code is sufficiently documented, and you are
welcome to ask more about its functionality -- it can do much more
than just computing these multiplication coefficients)

Hope this helps,
Dmitrii
http://www.ntu.edu.sg/home/dima/


On 13 February 2011 18:16, Pooja Singla <pooja at math.bgu.ac.il> wrote:
> Dear forum members,
>
>    I would like to know if Gap has some package to answer the following
> question:
>
> Let G be a finite group with a fixed subgroup H. Let Ha_1H, Ha_2H,..., Ha_nH
>
> be the complete set of representatives of H\G/H. Then is it possible to
> determine
> the multiplication tables Ha_i H.Ha_j H in terms of H a_k H (1 \leq i,j,k
> \leq n)?
>
> Any help in this regard will be very useful.
>
> Thanks,
> Pooja.
>
>
>
> --
>
>
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--
Dmitrii Pasechnik
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