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

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

2010-01-20

arxiv.org/abs/1001.3437v1

Advances in Mathematics, Volume 229, Issue 1, 15 (2012), 633--667

Mathematics

Algebraic Geometry

31 pages

Scientific paper

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.

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