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We give an explicit combinatorial description of the multiplicity as well asthe Hilbert function of the tangent cone at any point on a Schubert variety inthe symplectic Grassmannian.Comment: latex, 22 pages, 2 figures; for version 2: a few references added,  minor changes; for version 3: Final version to appear in TAMS, a few minor  (but potentially misleading) typos and erros correcte

Hilbert functions of points on Schubert varieties in the symplectic Grassmannian