On the shape of Bruhat intervals

Abstract

Let (W,S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and J a subset of S. Let $W^J$ denote the set of minimal coset representatives modulo the parabolic subgroup $W_J$. For w in $W^J$, let $f^{w,J}_{i}$ denote the number of elements of length i below w in Bruhat order on $W^J$ (notation simplified to $f^{w}_{i}$ in the case when J=S). We show that $f^{w,J}_{i}$ is less than or equal to $f^{w,J}_{j}$ when i < j and j is less than or equal to the length of w minus i. Furthermore, we express when an initial and final interval of the f's is symmetric around the middle in terms of Kazhdan-Lusztig polynomials. It is also shown that if W is finite then the sequence of f's cannot grow too rapidly. Som result mirroring our first result are obtaind, again in the finite case. The proofs rely for the most part on properties of the cohomology of Kac-Moody Schubert varieties.