Invariants and semi-invariants in the cohomology of the complement of a reflection arrangement

TL;DR

This paper constructs an explicit basis for the invariants in the cohomology of the complement of reflection arrangements, extending previous conjectures and simplifying related computations.

Contribution

It provides a natural basis for the invariants in the cohomology of reflection arrangement complements, generalizing prior conjectures to all finite complex reflection groups.

Findings

Explicit basis for invariants constructed

Extension of Felder and Veselov's conjecture to complex reflection groups

Simplification of existing cohomology calculations

Abstract

Suppose V is a finite dimensional, complex vector space, A is a finite set of codimension one subspaces of V, and G is a finite subgroup of the general linear group GL(V) that permutes the hyperplanes in A. In this paper we study invariants and semi-invariants in the graded QG-module H^*(M(A)), where M(A) denotes the complement in V of the hyperplanes in A and H^*( . ) denotes rational singular cohomology, in the case when A is a reflection arrangement and the pair (A,G) arises from a reflection coset. Our main result is the construction of an explicit, natural (from the point of view of Coxeter groups) basis of the space of invariants, H^*(M(A))^G. In addition to leading to a proof of the description of the space of invariants conjectured by Felder and Veselov for Coxeter groups that does not rely on computer calculations, this construction provides an extension of this description of the space of invariants to arbitrary finite, complex reflection groups. The main result also leads to simplifications of some cohomology computations of Lehrer, Callegaro-Marin, and Marin.