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.
As an application of @cite_40 , formulae for the multigraded Hilbert series of Kazhdan-Lusztig ideals were geometrically proved, where the multigrading comes from the torus action of the invertible diagonal matrices @math . While this theorem is actually used in a crucial way in the present paper, in general this Hilbert series does not help to directly compute multiplicity, because this torus action is not compatible with the dilation action. However, a Kazhdan-Lusztig ideal happens to already be homogeneous with respect to the standard grading that assigns each variable degree one, then it is automatic that it is also the ideal for its projectivized tangent cone, and one can deduce a formula for multiplicity from this Hilbert series (homogeneity is guaranteed if @math is @math -avoiding; see [pg. 25] Knutson:frob ). Moreover, it was explained that for the Grassmannian cases, one can always use the trick of to reduce to the homogeneous case. This gives an easy solution to the Grassmannian multiplicity problem, using Kazhdan-Lusztig ideals. Unfortunately, even for covexillary Schubert varieties, parabolic moving is ineffective for even some small examples. The approach of this paper avoids this issue, by using more direct arguments.
@cite_40
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Kazhdan-Lusztig ideals, a family of generalized determinantal ideals investigated in [Woo-Yong '08], provide an explicit choice of coordinates and equations encoding a neighbourhood of a torus-fixed point of a Schubert variety on a type A flag variety. Our main result is a Grobner basis for these ideals. This provides a single geometric setting to transparently explain the naturality of pipe dreams on the Rothe diagram of a permutation, and their appearance in: * combinatorial formulas [Fomin-Kirillov '94] for Schubert and Grothendieck polynomials of [Lascoux-Schutzenberger '82]; * the equivariant K-theory specialization formula of [Buch-Rimanyi '04]; and * a positive combinatorial formula for multiplicities of Schubert varieties in good cases, including those for which the associated Kazhdan-Lusztig ideal is homogeneous under the standard grading. Our results generalize (with alternate proofs) [Knutson-Miller '05]'s Grobner basis theorem for Schubert determinantal ideals and their geometric interpretation of the monomial positivity of Schubert polynomials. We also complement recent work of [Knutson '08,'09] on degenerations of Kazhdan-Lusztig varieties in general Lie type, as well as work of [Goldin '01] on equivariant localization and of [Lakshmibai-Weyman '90], [Rosenthal-Zelevinsky '01], and [Krattenthaler '01] on Grassmannian multiplicity formulas.