Representations of wreath products on cohomology of De Concini-Procesi compactifications

TL;DR

This paper derives formulas for the characters and Betti numbers of wreath product group actions on the cohomology of De Concini-Procesi compactifications, extending previous results and employing advanced combinatorial methods.

Contribution

It provides new character formulas and Betti number calculations for wreath product actions on these compactifications, generalizing prior work and introducing a novel application of tensor species theory.

Findings

Character formulas for wreath product actions on cohomology

Generalized Betti number formulas for compactifications

New proof of Lehrer's formula for wreath products

Abstract

The wreath product W(r,n) of the cyclic group of order r and the symmetric group S_n acts on the corresponding projective hyperplane complement, and on its wonderful compactification as defined by De Concini and Procesi. We give a formula for the characters of the representations of W(r,n) on the cohomology groups of this compactification, extending the result of Ginzburg and Kapranov in the r=1 case. As a corollary, we get a formula for the Betti numbers which generalizes the result of Yuzvinsky in the r=2 case. Our method involves applying to the nested-set stratification a generalization of Joyal's theory of tensor species, which includes a link between polynomial functors and plethysm for general r. We also give a new proof of Lehrer's formula for the representations of W(r,n) on the cohomology groups of the hyperplane complement.