Kazhdan-Lusztig polynomials and drift configurations

TL;DR

This paper establishes a combinatorial framework linking Kazhdan-Lusztig polynomials and local Schubert variety polynomials, proving conjectures for certain varieties and introducing drift configurations for new combinatorial rules.

Contribution

It proves conjectures relating Kazhdan-Lusztig and local Schubert polynomials for covexillary varieties and introduces drift configurations for combinatorial computation of these polynomials.

Findings

Proved nonnegativity and upper semicontinuity for specific cases.

Derived a combinatorial formula for local Schubert polynomials.

Established coefficient-wise inequalities between the polynomials.

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 $P_{v,w}(q)\preceq H_{v,w}(q)$.