[GAP Forum] Checking if permutation action has self-paired orbitals

[Dima Pasechnik]

Dear forum,

On Fri, Sep 20, 2013 at 07:17:09PM +0800, Robert Bailey wrote:
> Dear Frédéric,
>
> You may also be able to approach your problem by thinking of it in terms of coherent configurations and/or association schemes (an area which is unfortunately a terminological minefield), for which there are some useful GAP functions available.  The first is the "Elementary functions for association schemes on GAP" of Hanaki: see
> http://math.shinshu-u.ac.jp/~hanaki/as/gap/association_scheme.gap
>
> Also, Peter Cameron has some relevant GAP functions on his webpage, which work especially well for CCs obtained from permutation groups: see
> http://www.maths.qmul.ac.uk/~pjc/gapprogs.html
>
> For instance, Peter's programs include a function for testing if a group is generously transitive, i.e. every orbital is self-paired.

it is trivial to read the corresponding information off the output of
GRAPE's OrbitalGraphColadjMats (which works for transitive groups only,
but for intransitive groups you always have non-self-paired orbitals and
the representation is not multiplicity-free).

E.g.
gap> LoadPackage("grape");
gap> g:=PrimitiveGroup(7,3);;
gap> m:=OrbitalGraphColadjMats(g);;
gap> Display(m[2]);
[ [  0,  3,  0 ],
  [  0,  1,  2 ],
  [  1,  1,  1 ] ]

shows that the 2nd orbital is not self-paired, as
an orbital with the collapsed adjacency matrix A is self-paired iff there exists
k such that A[1][k]*A[k][1] is non-0, and the 1st of the matrices in the
output of OrbitalGraphColadjMats is always the identity matrix.

Also,
gap> m[2]*m[3]=m[3]*m[2];
true

shows that the representation is multiplicity-free (for this holds iff
the collapsed adjacency matrices commute).

Hope this helps,
Dmitrii

>
> I hope this is of some use to you!
>
> Regards,
> Robert.
>
> ----- Original Message -----
> From: Frederic Vanhove <fvanhove at cage.UGent.be>
> Date: Friday, September 20, 2013 6:02 am
> Subject: [GAP Forum] Checking if permutation action has self-paired orbitals
> To: "forum at gap-system.org" <forum at gap-system.org>
>
>
> > Dear forum,
> >
> > suppose you have a group acting on a set.  The orbitals are the orbits
> >
> > on ordered pairs of elements of that set, and their number
> > can be computed in GAP using
> > RankAction(groupname,setname);
> >
> > But I would like to know if these orbitals are self-paired, i.e. that
> >
> > (x1,x2) and (x2,x1) are always in the same orbit.
> > What is the easiest way to check this?
> >
> > More generally, I would like to check if the permutation character is
> > at
> > least multiplicity-free.
> >
> > Many thanks,
> > Kind regards,
> > Frédéric
> >
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>
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