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

Abstract

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.