Noncrossing partitions, Bruhat order and the cluster complex

TL;DR

This paper introduces new order relations on finite Coxeter groups, explores their properties, and connects them to noncrossing partitions and the cluster complex, providing new insights into Coxeter combinatorics.

Contribution

It defines two refined order relations on Coxeter groups and links their intervals to the cluster complex, enhancing understanding of Coxeter combinatorics.

Findings

Intervals can be enumerated via the cluster complex.

Revisits and unifies results like Chapoton triangles.

Establishes bijections between clusters and noncrossing partitions.

Abstract

We introduce two order relations on finite Coxeter groups which refine the absolute and the Bruhat order, and establish some of their main properties. In particular we study the restriction of these orders to noncrossing partitions and show that the intervals for these orders can be enumerated in terms of the cluster complex. The properties of our orders permit to revisit several results in Coxeter combinatorics, such as the Chapoton triangles and how they are related, the enumeration of reflections with full support, the bijections between clusters and noncrossing partitions.