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On the shape of Bruhat intervals
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-7497-2764
2009 (English)In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 170, no 2, p. 799-817Article in journal (Refereed) Published
##### Abstract [en]

Let (W, S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let J subset of S. Let W-J denote the set of minimal coset representatives modulo the parabolic subgroup W-J. For w is an element of W-J, let f(i)(w,J) denote the number of elements of length i below w in Bruhat order on W-J (with notation simplified to f(i)(w) in the case when W-J = W). We show that 0 <= i < j <= l(w)-i implies f(i)(w,J) <= f(j)(w,J). Also, the case of equalities f(i)(w) = f(l(w)-i)(w) for i = 1,..., k is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial P-e,P-w (q). We show that if W is finite then the number sequence f(0)(w), f(1)(w),... f(l(w))(w) cannot grow too rapidly. Further, in the finite case, for any given k >= 1 and any w is an element of W of sufficiently great length (with respect to k), we show f(l(w)-k)(w) >= f(l(w)-k+1)(w) >= ... >= f(l(w))(w). The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if (X) over bar (w) is a Schubert variety of dimension d = l(w), and lambda = c(1) (L) is an element of H-2 ((X) over bar (w)) is the restriction to (X) over bar (w) of the Chem class of an ample line bundle, then (lambda(k)) . : Hd-k((X) over bar (w)) -> Hd+k((X) over bar (w)) is injective for all k >= 0.

##### Place, publisher, year, edition, pages
2009. Vol. 170, no 2, p. 799-817
##### Keywords [en]
Coxeter group, Weyl group, Bruhat order, Schubert variety, l-adic, cohomology, intersection cohomology, Kazhdan-Lusztig polynomial, algebras, homology
##### Identifiers
ISI: 000271956100009Scopus ID: 2-s2.0-71449095628OAI: oai:DiVA.org:kth-18972DiVA, id: diva2:337019
##### Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2020-03-05Bibliographically approved

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