SB-Labelings, Distributivity, and Bruhat Order on Sortable Elements

TL;DR

This paper studies the structure of $eta$-sortable elements in Coxeter groups under Bruhat order, showing they form SB-lattices and identifying when these lattices are distributive, especially in certain 'coincidental' groups.

Contribution

It proves that all join-distributive lattices are SB-lattices and characterizes when $eta$-sortable element posets are distributive in specific Coxeter groups.

Findings

$eta$-sortable element posets are SB-lattices

These posets are distributive for groups $A_n$, $B_n$, $H_3$, and $I_2(k)$

Conjecture on forbidden orientations characterizing distributivity

Abstract

In this article, we investigate the set of $\gamma$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $\gamma\in W$, under Bruhat order, and we denote this poset by $\mathcal{B}_{\gamma}$. We show that this poset belongs to the class of SB-lattices recently introduced by Hersh and M\'esz\'aros, by proving a more general statement, namely that all join-distributive lattices are SB-lattices. The observation that $\mathcal{B}_{\gamma}$ is join-distributive is due to Armstrong. Subsequently, we investigate for which finite Coxeter groups $W$ and which Coxeter elements $\gamma\in W$ the lattice $\mathcal{B}_{\gamma}$ is in fact distributive. It turns out that this is the case for the "coincidental" Coxeter groups, namely the groups $A_{n},B_{n},H_{3}$ and $I_{2}(k)$. We conclude this article with a conjectural characteriziation of the Coxeter elements $\gamma$ of said groups for which $\mathcal{B}_{\gamma}$ is distributive in terms of forbidden orientations of the Coxeter diagram.