Noncrossing partitions and Bruhat order

TL;DR

This paper investigates the Bruhat order on noncrossing partitions related to Coxeter elements, establishing lattice structures, and exploring how changing Coxeter elements affects these properties and the associated bijections.

Contribution

It provides a criterion for Bruhat order on noncrossing partitions and analyzes how lattice structures vary with different Coxeter elements.

Findings

Bruhat order induces a lattice structure on noncrossing partitions for a fixed Coxeter element

Changing the Coxeter element can break the lattice property

Bijections between noncrossing partitions preserve the set of reflections

Abstract

We give a criterion for Bruhat order on noncrossing partitions corresponding to the Coxeter element $c=s_1 s_2\cdots s_n$. Using it we prove that the Bruhat order endows noncrossing partitions with a lattice structure. We then explain what happens if we change the Coxeter element; in that case the lattice property fails and we explain which order to consider to get the same lattice structure as for the Coxeter element $c$. In particular we get bijections between noncrossing partitions associated to distinct Coxeter elements, which fix the set of reflections.