On Multiplicities of Points on Schubert Varieties in Graßmannians II

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We prove a conjecture by Kreiman and Lakshmibai on a combinatorial description of multiplicities of points on Schubert varieties in Graßmannian in terms of certain sets of reflections in the corresponding Weyl group. The proof is accomplished by relating these sets of reflections to the author's previous combinatorial interpretation of these multiplicities in terms of non-intersecting lattice paths (Séminaire Lotharingien Combin. 45 (2001), Article B45c). Moreover, we provide a compact formula for the Hilbert series of the tangent cone to a Schubert variety in a Graßmannian assuming the truth of another conjecture of Kreiman and Lakshmibai.

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