Kazhdan–Lusztig polynomials and drift configurations

Abstract

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