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

We study Hilbert-Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Gröbner degenerations of the Kazhdan-Lusztig ideal. In the covexillary case, we give a manifestly positive combinatorial rule for multiplicity by establishing (with a Gröbner basis) a reduced 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 and Weyman (1990) [26], Rosenthal and Zelevinsky (2001) [37], Krattenthaler (2001) [22], Kodiyalam and Raghavan (2003) [21], Kreiman and Lakshmibai (2004) [24], Ikeda and Naruse (2009) [13] and Woo and Yong (2009) [40]. We suggest extensions of our methodology to the general case.