[GAP Forum] natural representation and orbits

[Bill Allombert]

On Sun, Oct 18, 2015 at 08:49:15AM -0600, Alexander Hulpke wrote:
> 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.

Thanks for your very useful answer.  Maybe I should give some motivation:

Let K be a number field, L its Galois closure over Q (the rational field), and
G = Gal(L/Q). The Dedekind Zeta function of K is equal to the Artin L function
associated to the natural representation of G (seen as acting on the complex
embedding of K). (the Artin L function is an arithmetic object functiorialy
attached to representations of Galois groups of numbers field).

Thus it factors as a product of Artin L functions associated to the irreducible
representations that occurs in the natural representation.

Computing this factorization is very important for computing the Dedekind Zeta
function.

However computing the Galois group G is difficult (This is the GAP function
GaloisType). But some property of G are easy to compute (e.g. primitiveness,
and the orbits under the stabilisator of a point).

So any trick which can allow to compute the factorization without computing
the exact Galois group is useful.

Cheers,
Bill