#### Vol. 5, No. 5, 2011

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors' Addresses Editors' Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)
Kazhdan–Lusztig polynomials and drift configurations

### Li Li and Alexander Yong

Vol. 5 (2011), No. 5, 595–626
##### 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 ${P}_{v,w}\left(q\right)\preccurlyeq {H}_{v,w}\left(q\right)$.

##### Keywords
Kazhdan–Lusztig polynomials, Hilbert series, Schubert varieties
##### Mathematical Subject Classification 2000
Primary: 14M15
Secondary: 05E15, 20F55