Eigenspace arrangements of reflection groups

TL;DR

This paper investigates the topology and homology of eigenspace arrangements of complex reflection groups, extending known models and establishing shellability and K(pi,1) properties for these arrangements.

Contribution

It generalizes the intersection lattice framework to arbitrary eigenvalues, proves shellability for many groups, and introduces balanced partition posets for symmetric groups.

Findings

All upper intervals are geometric lattices.

Shellability is established for G(m,p,n) and most exceptional groups.

Characterization of when eigenspaces form K(pi,1) arrangements.

Abstract

The lattice of intersections of reflecting hyperplanes of a complex reflection group W may be considered as the poset of 1-eigenspaces of the elements of W. In this paper we replace 1 with an arbitrary eigenvalue and study the topology and homology representation of the resulting poset. After posing the main question of whether this poset is shellable, we show that all its upper intervals are geometric lattices, and then answer the question in the affirmative for the infinite family G(m,p,n) of complex reflection groups, and the first 31 of the 34 exceptional groups, by constructing CL-shellings. In addition, we completely determine when these eigenspaces of W form a K(pi,1) (resp. free) arrangement. For the symmetric group, we also extend the combinatorial model available for its intersection lattice to all other eigenvalues by introducing "balanced partition posets", presented as particular upper order ideals of Dowling lattices, study the representation afforded by the top (co)homology group, and give a simple map to the posets of pointed d-divisible partitions.