Noncrossing partitions and Bruhat order

TL;DR

This paper establishes a distributive lattice structure on noncrossing partitions in type A_n under Bruhat order, extending bijections between noncrossing partitions for different Coxeter elements and connecting to algebraic bases.

Contribution

It proves the distributive lattice structure of noncrossing partitions under Bruhat order and extends bijections to other Coxeter elements via canonical factorizations.

Findings

Bruhat order restriction forms a distributive lattice

Bijections between noncrossing partitions for different Coxeter elements

Connections to Temperley-Lieb algebra and dual braid monoid

Abstract

We prove that the restriction of Bruhat order to noncrossing partitions in type $A_n$ for the Coxeter element $c=s_1s_2 ...s_n$ forms a distributive lattice isomorphic to the order ideals of the root poset ordered by inclusion. Motivated by the change-of-basis from the graphical basis of the Temperley-Lieb algebra to the image of the simple elements of the dual braid monoid, we extend this bijection to other Coxeter elements using certain canonical factorizations. In particular, we give new bijections---fixing the set of reflections---between noncrossing partitions associated to distinct Coxeter elements.