Descent Systems for Bruhat Posets

Let (W,S) be a finite Weyl group and let w ∈ W . It is widely appreciated that the descent set D(w) = {s ∈ S | l(ws) < l(w)} determines a very large and important chapter in the study of Coxeter groups. In this paper we generalize some of those results to the situation of the Bruhat poset W J where J ⊂ S. Our main results here include the identification of a certain subset S ⊂ W J that convincingly plays the role of S ⊂ W , at least from the point of view of descent sets and related geometry. The point here is to use this resulting descent system (W J , S) to explicitly encode some of the geometry and combinatorics that is intrinsic to the poset W J . In particular, we arrive at the notion of an augmented poset, and we identify the combinatorially smooth subsets J ⊂ S that have special geometric significance in terms of a certain corresponding torus embedding X(J).