Kazhdan-Lusztig polynomials and drift configurations

Abstract

The coefficients of the Kazhdan-Lusztig polynomials $P_{v,w}(q)$ are nonnegative integers that are upper semicontinuous on 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 \emph{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 $P_{v,w}(q)⪯H_{v,w}(q)$.

26 pages. To appear in Algebra & Number Theory

References in corpus (5)

• Governing Singularities of Schubert Varieties
• Frobenius splitting, point-counting, and degeneration
• A Gröbner basis for Kazhdan-Lusztig ideals
• Quasisymmetric functions and Kazhdan-Lusztig polynomials
• Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties

Cited by in corpus (1)

• Patch ideals and Peterson varieties