[GAP Forum] natural representation and orbits
Bill Allombert
Bill.Allombert at math.u-bordeaux.fr
Sun Oct 18 10:15:00 BST 2015
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.
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