[GAP Forum] natural representation and orbits

[Bill Allombert]

Dear GAP forum,

Let G be a primitive transitive subgroup of S_n.
I am interested by the links between:
1) the lengths of the orbits of {1,...,n} under the action of the stabilisator
of 1 by G.
2) the degrees of the irreducible representations occuring in the natural
representation of G.

I wrote the GAP code below that shows the two entities above are often equal.
(I had to filter out representation of degree 1 because Orbits() does not 
return orbits of length 1).

orb:=function(T)
 return List(Orbits(Stabilizer(T,1)),Length);
end;

irr:=function(T)
  return Filtered(List(ConstituentsOfCharacter(NaturalCharacter(T)),DegreeOfCharacter),x->x>1);
end;

nat:=function(n)
  local L;
  L:=AllTransitiveGroups(NrMovedPoints,n,IsPrimitive,true);
  return Filtered(L,T->orb(T) <> irr(T));
end;
for i in [2..17] do Print(i, ":", nat(i),"\n"); od;

This the list of counter-examples for n<=17

2:[  ]
3:[  ]
4:[  ]
5:[  ]
6:[  ]
7:[  ]
8:[  ]
9:[  ]
10:[ A_5(10), S_5(10d) ]
11:[  ]
12:[  ]
13:[  ]
14:[  ]
15:[ A_6(15), S_6(15) ]
16:[ t16n708, t16n711, t16n1030, t16n1034, t16n1294 ]
17:[  ]

So I would be very much interested by pointers to an explanation of this
phenomena.

Sorry that my question is more about group theory than GAP itself, but I would
have had trouble expressing it without GAP.

Cheers,
Bill.