Topology of eigenspace posets for imprimitive reflection groups

TL;DR

This paper investigates the structure of eigenspace posets in imprimitive reflection groups, revealing their isomorphism to Dowling lattice subposets and establishing their Cohen-Macaulay property and homology dimensions.

Contribution

It introduces a novel connection between eigenspace posets of imprimitive reflection groups and Dowling lattices, proving their Cohen-Macaulayness and calculating top homology dimensions.

Findings

Posets are isomorphic to subposets of Dowling lattices.

Eigenspace posets are Cohen-Macaulay.

Dimensions of their top homology are determined.

Abstract

This paper studies the poset of eigenspaces of elements of an imprimitive unitary reflection group, for a fixed eigenvalue, ordered by the reverse of inclusion. The study of this poset is suggested by the eigenspace theory of Springer and Lehrer. The posets are shown to be isomorphic to certain subposets of Dowling lattices (the `d-divisible, k-evenly coloured Dowling lattices'). This enables us to prove that these posets are Cohen-Macaulay, and to determine the dimension of their top homology.