[GAP Forum] Hilbert series

[Thomas Breuer]

Dear GAP Forum,

Won Kyu Park wrote

> I want to get hilbert series of permutation group G
> 
> gap> G;
> Group([ (2,3)(4,5)(7,8)(11,12)(16,17),
> (1,3,2)(4,5,6)(7,8,9)(11,12,13)(16,17,18),
> (4,7)(5,8)(6,9)(14,15)(19,20),
>   (4,7,11)(5,8,12)(6,9,13)(10,15,14)(19,20,21) ])
> gap> IsPermGroup(G);
> true
> gap> IG:=Irr(G);;
> gap> List([1..Size(IG)],i->MolienSeries(IG[i]));
> [ ( 1 ) / ( (1-z) ), ( 1 ) / ( (1-z^2) ), ( 1 ) / ( (1-z^2) ), ( 1 ) /
> ( (1-z^2) ), ( 1 ) / ( (1-z^3)*(1-z^2) ),
>   ( 1 ) / ( (1-z^6)*(1-z^2) ), ( 1 ) / ( (1-z^6)*(1-z^2) ), ( 1 ) / (
> (1-z^3)*(1-z^2) ),
>   ( 1-z+z^4-z^7+z^8 ) / ( (1-z^6)*(1-z^3)*(1-z^2)*(1-z) ) ]
> 
> from http://www.math.colostate.edu/manuals/magma/htmlhelp/text419.html
> "If G is a permutation group, the Molien series always exists and
> equals the Hilbert series of the invariant ring of G for any field."
> 
> how can i get the hilbert series from the output of GAP ?

The output of `MolienSeries' is a rational function f(z), say.

GAP can deal with the infinite series \sum_{i=0}^{\infty} a_i z^i
represented by this rational function only in the sense
that one can compute the coefficients a_i for given i.
In the above example, this could be done as follows
(see also the section ``Molien Series'' in the GAP Reference Manual).

    gap> mols:= List( Irr( G ), MolienSeries );;
    gap> mols[5];
    ( 1 ) / ( (1-z^3)*(1-z^2) )
    gap> List( [ 0 .. 20 ], i -> ValueMolienSeries( mols[5], i ) ); 
    [ 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4 ]

So a_0 = a_2 = a_3 = a_4 = a_5 = a_7 = 1, a_1 = 0, etc.

Besides that, the `MolienSeriesInfo' value of a Molien series object
contains information how the Molien series can be written as a sum
of terms of the form g(z)/(1-z^r)^s, for suitable polynomials g(z);
from this, one could derive a more or less closed form
for the coefficients a_i,
either by hand or using a computer system (other than GAP) that supports
such manipulations.

All the best,
Thomas Breuer