Descent Systems for Bruhat Posets

Abstract

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 WJ where J⊂eq S. Our main results here include the identification of a certain subset SJ⊂eq WJ that convincingly plays the role of S⊂eq W, at least from the point of view of descent sets and related geometry. The point here is to use this resulting descent system (WJ,SJ) to explicitly encode some of the geometry and combinatorics that is intrinsic to the poset WJ. In particular, we arrive at the notion of an augmented poset, and we identify the combinatorially smooth subsets J⊂eq S that have special geometric significance in terms of a certain corresponding torus embedding X(J). The theory of J-irreducible monoids provides an essential tool in arriving at our main results.