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

[Thomas Breuer]

Dear Frederic,

you wrote to the GAP 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.

In general, the question whether two pairs are in the same orbit
can be checked with 'RepresentativeAction'.
Here is an example:

    gap> g:= SymmetricGroup( 3 );;
    gap> RepresentativeAction( g, [ 1, 2 ], [ 1, 3 ], OnPairs );
    (2,3)
    gap> RepresentativeAction( g, [ 1, 1 ], [ 1, 3 ], OnPairs );
    fail

For that, you need representatives of the orbitals.
If your group acts transitively then you can get them
from orbit representatives of a point stabilizer.
(See the GAP Reference Manual for more details.)


Concerning the question about the permutation character of the action,
one can compute this character explicitly, and if one knows the
irreducible characters of the group then also the decomposition of the
permutation character into irreducibles can be computed.

However, if the groups in question are large then this is perhaps not
the best approach; if the character tables of these groups are available
in GAP's library of character tables then it might be possible to decide
questions about transitive permutation characters with purely character
theoretic methods.
For relevant GAP functions, see section ``Possible Permutation Characters''
in the GAP Reference Manual.
For some examples, see

    http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/htm/ctblpope.htm

All the best,
Thomas


On Fri, Sep 20, 2013 at 05:30:01PM +0900, Frederic Vanhove wrote:
> 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