On the shape of Bruhat intervals

Abstract

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 $$ 0\le i\lt j\le \ell (w)-i \quad\hbox{implies}\quad f^{w\smash{,J}}_{i} \le f^{w\smash{,J}}_{j}.$$ Also, the case of equalities $\smash{f^{w}_{i} = f^{w}_{\ell(w)-i}}$ for $i=1, \ldots,k$ is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial $P_{e,w}(q)$.

We show that if $W$ is finite then the number sequence $\smash{f^{w}_{0}, f^{w}_{1}, \ldots, f^{w}_{\ell (w)}}$ cannot grow too rapidly. Further, in the finite case, for any given $k\ge 1$ and any $w\in W$ of sufficiently great length (with respect to $k$), we show $$ f^{w}_{\ell(w)-k} \ge f^{w}_{\ell(w)-k+1} \ge\cdots \ge f^{w}_{\ell (w)}.$$

The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if $\mskip3mu\overline{\mskip-3mu X} _w$ is a Schubert variety of dimension $d=\ell (w)$, and $\lambda=c_1 (\mathscr L)\in H^2 (\mskip3mu\overline{\mskip-3mu X} _w)$ is the restriction to $\mskip3mu\overline{\mskip-3mu X} _w$ of the Chern class of an ample line bundle, then \[(\lambda^k)\,\cdot \,{} : H^{d-k}(\mskip3mu\overline{\mskip-3mu X} _w) \rightarrow H^{d+k}(\mskip3mu\overline{\mskip-3mu X} _w) \] is injective for all $k\ge 0$.