Reflection arrangements and ribbon representations

TL;DR

This paper extends the connection between reflection arrangements and ribbon representations from type A to all real and Shephard complex reflection groups, unifying previous results and highlighting geometric aspects.

Contribution

It generalizes the homotopy equivalence and top homology identification from type A to all real and Shephard reflection groups, broadening the scope of prior work.

Findings

Unified the homotopy equivalence for all reflection groups.

Extended top homology identification to complex Shephard groups.

Strengthened the geometric understanding of reflection arrangements.

Abstract

Ehrenborg and Jung recently related the order complex for the lattice of d-divisible partitions with the simplicial complex of pointed ordered set partitions via a homotopy equivalence. The latter has top homology naturally identified as a Specht module. Their work unifies that of Calderbank, Hanlon, Robinson, and Wachs. By focusing on the underlying geometry, we strengthen and extend these results from type A to all real reflection groups and the complex reflection groups known as Shephard groups.