Exponential formulas for models of complex reflection groups
Abstract
In this paper we find some exponential formulas for the Betti numbers of the De Concini-Procesi minimal wonderful models YG(r,p,n) associated to the complex reflection groups G(r,p,n). Our formulas are different from the ones already known in the literature: they are obtained by a new combinatorial encoding of the elements of a basis of the cohomology by means of set partitions with weights and exponents. We also point out that a similar combinatorial encoding can be used to describe the faces of the real spherical wonderful models of type An-1=G(1,1,n), Bn=G(2,1,n) and Dn=G(2,2,n). This provides exponential formulas for the f-vectors of the associated nestohedra: the Stasheff's associahedra (in this case closed formulas are well known) and the graph associahedra of type Dn.