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Kazhdan-Lusztig polynomials and drift configurations

Abstract

The coefficients of the Kazhdan-Lusztig polynomials Pv,w(q)P_{v,w}(q) are nonnegative integers that are upper semicontinuous on Bruhat order. Conjecturally, the same properties hold for hh-polynomials Hv,w(q)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 Hv,w(q)H_{v,w}(q). We introduce \emph{drift configurations} to formulate a new and compatible combinatorial rule for Pv,w(q)P_{v,w}(q). From our rules we deduce, for these cases, the coefficient-wise inequality Pv,w(q)Hv,w(q)P_{v,w}(q)\preceq H_{v,w}(q).Comment: 26 pages. To appear in Algebra & Number Theor

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