The sorting order on a Coxeter group

Let (W,S) be an arbitrary Coxeter system. For each sequence ω = (ω1, ω2, . . .) ∈ S∗ in the generators we define a partial order—called the ω-sorting order—on the set of group elements Wω ⊆ W that occur as finite subwords of ω. We show that the ω-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and strong Bruhat orders on the group. Moreover, the ω-sorting order is a “maximal lattice” in the sense that the addition of any collection of edges from the Bruhat order results in a nonlattice. Along the way we define a class of structures called supersolvable antimatroids and we show that these are equivalent to the class of supersolvable join-distributive lattices.