Descent Systems for Bruhat Posets

Let $(W,S)$ be a finite Weyl group and let $w\in W$. It is widely appreciated that the descent set D(w)=\{s\in 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 $J\subseteq S$. Our main results here include the identification of a certain subset $S^J\subseteq W^J$ that convincingly plays the role of $S\subseteq W$, at least from the point of view of descent sets and related geometry. The point here is to use this resulting {\em descent system} $(W^J,S^J)$ 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 {\em augmented poset}, and we identify the {\em combinatorially smooth} subsets $J\subseteq S$ that have special geometric significance in terms of a certain corresponding torus embedding $X(J)$. The theory of $\mathscr{J}$-irreducible monoids provides an essential tool in arriving at our main results.