Topology of eigenspace posets for imprimitive reflection groups

Justin Koonin

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.

Keywords: Poset topology, unitary reflection groups, imprimitive reflection groups, Dowling lattices, exponential Dowling structures.

AMS Subject Classification: Primary 05E45; secondary 20F55.