[GAP Forum] natural representation and orbits
Alexander Hulpke
hulpke at fastmail.fm
Sun Oct 18 15:49:15 BST 2015
Dear Bill,
> 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.
Frobenius reciprocity and the fact that the permutation character (=natural character) is the induced trivial representation of the point stabilizer show that the permutation character has inner product m with itself, where m is the number of orbits of the point stabilizer.
Thus the observation is true for doubly transitive groups: the permutation character has the form 1+chi with chi irreducible, so deg chi must be | \Omega |-1, thus proving the statement.
It also is true (for trivial reasons) for regular groups and (an ad-hoc observation) dihedral groups of prime degree.
This covers all but 22 of the primitive groups of degree up to 17. Of these, 7 are Frobenius groups for which I think again an ad-hoc argument works, leaving 15 groups, of which 9 fail (and 6 pass) the conjecture. So chances are about 50%.
But 50% is still somewhat surprising. I suspect the reason is that character degrees must divide up the group order and length of stabilizer orbits divide the stabilizer order. Trying to write a smallish (in this case <=17) number as sum of a few divisors leaves open only a few possibilities, making it likely that the same numbers are involved.
Best,
Alexander
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