SB-lattices, distributivity, and Bruhat order on sortable elements.

Henri Mühle¹

¹Université Paris Diderot LIAFA

The electronic journal of combinatorics, Tome 22 (2015) no. 2

Résumé

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 Mészáros, 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.

Détail

DOI : 10.37236/4942

Classification : 06A07, 20F55, 06D75
Mots-clés : SB-labelings, Möbius functions, crosscut complexes, distributive lattices, join-distributive lattices, antimatroids, Bruhat order, sortable elements, Coxeter groups