On Reflection Orders Compatible with a Coxeter Element

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 Björner and Wachs.

This article was withdrawn, since the generalized statement that any compatible order below some reflection group element in absolute order is a recursive atom order is wrong. A counterexample is for instance the absolute order interval between the identity and the longest element in $H_3$. The statement for Coxeter elements is probably true. Comments welcome