Kazhdan–Lusztig polynomials and drift configurations

Abstract

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).