Descent systems for Bruhat posets

Renner, Lex E.

Journal of Algebraic Combinatorics, Tome 29 (2009) no. 4, pp. 413-435.

Résumé

Summary: Let $( W, S)$ be a finite Weyl group and let $w\in W$. It is widely appreciated that the descent set $D( w)={ s$ Ĩ $S | l( ws) l( w)} D(w)$={s$\in S\mid l(ws)$ $(w)$} 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 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 augmented poset, and we identify the combinatorially smooth subsets $J\subseteq S$ 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.

Détail

Keywords: keywords Weyl group, Bruhat poset, descent system, augmented poset, $J$-irreducible monoid, rationally smooth