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⊆S. Let WJ denote the set of minimal coset representatives modulo the parabolic subgroup WJ. For w ε WJ, let fiω,J denote the number of elements of length i below w in Bruhat order on WJ (with notation simplied to fiω in the case when WJ = W). We show that Also, the case of equalities fiω = f(ω)-iω for i = 1,...,k is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial Pe,ω(q). We show that if W is finite then the number sequence f0ω, f1ω,...,f(ω)ω cannot grow too rapidly. Further, in the finite case, for any given k ≥ 1 and any ω ε W of sufficiently great length (with respect to k), we show The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if X̄ω is a Schubert variety of dimension d = (ω), and λ=c1(L) ε H2(Xω) is the restriction to X̄ω of the Chern class of an ample line bundle, then is injective for all k>0.