Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties

TL;DR

This paper investigates the Hilbert-Samuel multiplicity of points in Schubert varieties using Groebner degenerations, providing combinatorial rules and formulas, especially for covexillary and Grassmannian cases.

Contribution

It introduces a new combinatorial rule for multiplicity in covexillary Schubert varieties via Groebner degenerations and extends formulas to Grassmannian cases.

Findings

Provides a positive combinatorial rule for multiplicity.

Establishes a reduced, equidimensional limit with a shellable simplicial complex.

Offers formulas for Hilbert series of local rings.

Abstract

We study Hilbert-Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Groebner degenerations of the Kazhdan-Lusztig ideal. In the covexillary case, we give a positive combinatorial rule for multiplicity by establishing (with a Groebner basis) a reduced and equidimensional limit whose Stanley-Reisner simplicial complex is homeomorphic to a shellable ball or sphere. We show that multiplicity counts the number of facets of this complex. We also obtain a formula for the Hilbert series of the local ring. In particular, our work gives a multiplicity rule for Grassmannian Schubert varieties, providing alternative statements and proofs to formulae of [Lakshmibai-Weyman '90], [Rosenthal-Zelevinsky '01], [Krattenthaler '01], [Kreiman-Lakshmibai '04] and [Woo-Yong '09]. We suggest extensions of our methodology to the general case.