Exponential formulas for models of complex reflection groups

TL;DR

This paper introduces new exponential formulas for Betti numbers of complex reflection group models using combinatorial set partitions, also extending to real spherical models and associated polyhedral structures.

Contribution

It presents novel combinatorial encoding methods for cohomology bases, leading to new exponential formulas for Betti numbers and face counts of related polytopes.

Findings

New exponential formulas for Betti numbers of complex reflection group models.

Combinatorial encoding of cohomology basis elements via weighted set partitions.

Extension of formulas to faces of real spherical models and associated nestohedra.

Abstract

In this paper we find some exponential formulas for the Betti numbers of the De Concini-Procesi minimal wonderful models Y_{G(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 A_{n-1}=G(1,1,n), B_n=G(2,1,n) and D_n=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 D_n.