Homology representations of unitary reflection groups

TL;DR

This paper explores the representation theory of unitary reflection groups through the homology of eigenspace posets, extending previous work and unifying known representations into a natural eigenvalue-parameterized family.

Contribution

It introduces a new framework connecting eigenspace posets with the representation theory of unitary reflection groups using poset extension theory.

Findings

Representations of unitary reflection groups are placed into a natural family parameterized by eigenvalue.

The study extends previous work on eigenspace posets and their homology.

A new connection between hyperplane intersection lattices and group representations is established.

Abstract

This paper continues the study of the poset of eigenspaces of elements of a unitary reflection group (for a fixed eigenvalue), which was commenced in [6] and [5]. The emphasis in this paper is on the representation theory of unitary reflection groups. The main tool is the theory of poset extensions due to Segev and Webb ([16]). The new results place the well-known representations of unitary reflection groups on the top homology of the lattice of intersections of hyperplanes into a natural family, parameterised by eigenvalue.