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 _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
