Vol. 5, No. 5, 2011

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Kazhdan–Lusztig polynomials and drift configurations

Li Li and Alexander Yong

Vol. 5 (2011), No. 5, 595–626

The coefficients of the Kazhdan–Lusztig polynomials Pv,w(q) are nonnegative integers that are upper semicontinuous relative to Bruhat order. Conjecturally, the same properties hold for h-polynomials Hv,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 Hv,w(q). We introduce drift configurations to formulate a new and compatible combinatorial rule for Pv,w(q). From our rules we deduce, for these cases, the coefficient-wise inequality Pv,w(q) Hv,w(q).

Kazhdan–Lusztig polynomials, Hilbert series, Schubert varieties
Mathematical Subject Classification 2000
Primary: 14M15
Secondary: 05E15, 20F55
Received: 19 June 2010
Revised: 22 August 2010
Accepted: 1 October 2010
Published: 23 January 2012
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