Kazhdan–Lusztig polynomials and drift configurations

Li, Li; Yong, Alexander

doi:10.2140/ant.2011.5.595

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Kazhdan–Lusztig polynomials and drift configurations

Li Li and Alexander Yong

Vol. 5 (2011), No. 5, 595–626

DOI: 10.2140/ant.2011.5.595

Abstract

The coefficients of the Kazhdan–Lusztig polynomials $P_{v, w} (q)$ are nonnegative integers that are upper semicontinuous relative to Bruhat order. Conjecturally, the same properties hold for $h$ -polynomials $H_{v, w} (q)$ of local rings of Schubert varieties. This suggests a parallel between the two families of polynomials. We prove our conjectures for Grassmannians, and more generally, covexillary Schubert varieties in complete flag varieties, by deriving a combinatorial formula for $H_{v, w} (q)$ . We introduce drift configurations to formulate a new and compatible combinatorial rule for $P_{v, w} (q)$ . From our rules we deduce, for these cases, the coefficient-wise inequality $P_{v, w} (q) ≼ H_{v, w} (q)$ .

Keywords

Kazhdan–Lusztig polynomials, Hilbert series, Schubert varieties

Mathematical Subject Classification 2000

Primary: 14M15

Secondary: 05E15, 20F55

Milestones

Received: 19 June 2010

Revised: 22 August 2010

Accepted: 1 October 2010

Published: 23 January 2012

Authors

Li Li
Department of Mathematics and Statistics Oakland University Rochester, MI 48309 United States

Alexander Yong
Department of Mathematics University of Illinois at Urbana–Champaign Urbana, IL 61801 United States

Vol. 5, No. 5, 2011