On Reflection Orders Compatible with a Coxeter Element

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 $γ$ in any well-generated complex reflection group admits a $γ$-compatible reflection order, the lexicographic shellability follows from a well-known result due to Björner and Wachs.

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