SB-Labelings, Distributivity and Bruhat Order on Sortable Elements

In this article, we investigate the set of γ-sortable elements, associated with a Coxeter group W and a Coxeter element γ ∈ W , under Bruhat order, and we denote this poset by Bγ . 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 Bγ is join-distributive is due to Armstrong. Subsequently, we investigate for which finite Coxeter groups W and which Coxeter elements γ ∈ W the lattice Bγ is in fact distributive. It turns out that this is the case for the “coincidental” Coxeter groups, namely the groups An, Bn, H3, and I2(k). We conclude this article with a conjectural characteriziation of the Coxeter elements γ of said groups for which Bγ is distributive in terms of forbidden orientations of the Coxeter diagram.