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) , Rosenthal and Zelevinsky (2001) , Krattenthaler (2001) , Kodiyalam and Raghavan (2003) , Kreiman and Lakshmibai (2004) , Ikeda and Naruse (2009)  and Woo and Yong (2009) . We suggest extensions of our methodology to the general case. © 2011 Elsevier Inc..
Li, L., & Yong, A. (2012). Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties. Advances in Mathematics, 229(1), 633–667. https://doi.org/10.1016/j.aim.2011.09.010