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arXiv · 1405.3522

On Reflection Orders Compatible with a Coxeter Element

Henri Mühle
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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.

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Henri Mühle. 2014-05-14. On Reflection Orders Compatible with a Coxeter Element. https://arxiv.org/abs/1405.3522

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