On the shape of Bruhat intervals

Anders Björner ¹ ; Torsten Ekedahl²

¹ Department of Mathematics<br/>Royal Institute of Technology<br/>100 44 Stockholm<br/>Sweden
² Department of Mathematics<br/>Stockholm University<br/>106 91 Stockholm<br/>Sweden

Annals of mathematics, Tome 170 (2009) no. 2, pp. 799-817

Résumé

Let $(W,S)$ be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let $J\subseteq S$. Let $W^J$ denote the set of minimal coset representatives modulo the parabolic subgroup $W_J$. For $w\in W^J$, let $f^{w\smash{,J}}_{i}$ denote the number of elements of length $i$ below $w$ in Bruhat order on $W^J$ (with notation simplified to $f^{w}_{i}$ in the case when $W^J=W$). We show that $$ 0\le i\lt j\le \ell (w)-i \quad\hbox{implies}\quad f^{w\smash{,J}}_{i} \le f^{w\smash{,J}}_{j}. \end{displaymath} Also, the case of equalities $\smashf^w_i = f^w_\ell(w)-i$ for $i=1, \ldots,k$ is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial $P_e,w(q)$.

Détail

DOI : 10.4007/annals.2009.170.799