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.
While this paper focuses on type @math , our results should have analogues for other Lie types. Recent papers of A. Knutson @cite_13 @cite_34 point the way towards coordinates and equations for Kazhdan-Lusztig varieties. His papers also explain how to iteratively degenerate these varieties, although the degenerations he considers are not directly applicable in general to the multiplicity problem, since they do not degenerate the projectivized tangent cone. Finally, we remark that the notion of covexillary for type @math has already been examined in a paper by S. Billey and T. K. Lam @cite_7 .
@cite_34 @cite_13 @cite_7
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We study the intersections of general Schubert varieties X_w with permuted big cells, and give an inductive degeneration of each such "Schubert patch" to a Stanley-Reisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using the results of [Knutson 2007] on automatically reduced degenerations, or through more standard cohomology-vanishing arguments. The underlying simplicial complex of the Stanley-Reisner scheme is a subword complex, as introduced for slightly different purposes in [Knutson-Miller 2004], and is homeomorphic to a ball. This gives a new proof of the Andersen-Jantzen-Soergel Billey and Graham Willems formulae for restrictions of equivariant Schubert classes to fixed points. In analogy with the symmetric group, we define the vexillary elements in the hyperoctahedral group to be those for which the Stanley 1 symmetric function is a single Schur Q-function. We show that the vexillary elements can be again determined by pattern avoidance conditions. These results can be extended to include the root systems of types A, B, C, and D. Finally, we give an algorithm for multiplication of Schur Q -functions with a superfied Schur function and a method for determining the shape of a vexillary signed permutation using jeu de taquin.