Kazhdan-lusztig polynomials and drift configurations

The coefficients of the Kazhdan-Lusztig polynomials P_v;_w.q/ are nonnegative integers that are upper semicontinuous relative to 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 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 P_v;_w.q/≤H_v;_w.q/.