On Reflection Orders Compatible with a Coxeter Element

TL;DR

This paper proves that the lattice of noncrossing partitions for well-generated complex reflection groups is lexicographically shellable, extending known results from Coxeter groups using a simple, uniform proof approach.

Contribution

It provides a uniform proof that the noncrossing partition lattice is lexicographically shellable for all well-generated complex reflection groups, not just Coxeter groups.

Findings

Noncrossing partition lattice is lexicographically shellable for complex reflection groups.

Every x-compatible reflection order is a recursive atom order.

The result extends shellability from Coxeter groups to all well-generated complex reflection groups.

Abstract

In this article we give a simple, almost uniform proof that the lattice of noncrossing partitions associated with a well-generated complex reflection group is lexicographically shellable. So far a uniform proof is available only for Coxeter groups. In particular we show that, for any complex reflection group $W$ and any element $x\in W$, every $x$-compatible reflection order is a recursive atom order of the corresponding interval in absolute order. Since any Coxeter element $\gamma$ in any well-generated complex reflection group admits a $\gamma$-compatible reflection order, the lexicographic shellability follows from a well-known result due to Bj\"orner and Wachs.