References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt145",{id:"formSmash:upper:j_idt145",widgetVar:"widget_formSmash_upper_j_idt145",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt146_j_idt148",{id:"formSmash:upper:j_idt146:j_idt148",widgetVar:"widget_formSmash_upper_j_idt146_j_idt148",target:"formSmash:upper:j_idt146:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

On the shape of Bruhat intervalsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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var panelSome = $(PrimeFaces.escapeClientId("formSmash:some"));
var panelAll = $(PrimeFaces.escapeClientId("formSmash:all"));
panelAll.toggle();
toggleList(panelSome.get(0).childNodes, panelAll);
toggleList(panelAll.get(0).childNodes, panelAll);
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/*Toggling the list of authorPanel nodes according to the toggling of the closeable second panel */
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2009 (English)In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 170, no 2, 799-817 p.Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2009. Vol. 170, no 2, 799-817 p.
##### Keyword [en]

Coxeter group, Weyl group, Bruhat order, Schubert variety, l-adic, cohomology, intersection cohomology, Kazhdan-Lusztig polynomial, algebras, homology
##### Identifiers

URN: urn:nbn:se:kth:diva-18972ISI: 000271956100009ScopusID: 2-s2.0-71449095628OAI: oai:DiVA.org:kth-18972DiVA: diva2:337019
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt375",{id:"formSmash:j_idt375",widgetVar:"widget_formSmash_j_idt375",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt381",{id:"formSmash:j_idt381",widgetVar:"widget_formSmash_j_idt381",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt387",{id:"formSmash:j_idt387",widgetVar:"widget_formSmash_j_idt387",multiple:true});
##### Note

QC 20100525Available from: 2010-08-05 Created: 2010-08-05Bibliographically approved

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.

References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1080",{id:"formSmash:lower:j_idt1080",widgetVar:"widget_formSmash_lower_j_idt1080",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1081_j_idt1083",{id:"formSmash:lower:j_idt1081:j_idt1083",widgetVar:"widget_formSmash_lower_j_idt1081_j_idt1083",target:"formSmash:lower:j_idt1081:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});