Geodesic
Hilbert functions of points on Schubert varieties in orthogonal grassmannians
Journal of Algebraic Combinatorics, Tome 31 (2010) no. 3, pp. 355-409.

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Summary: Given a point on a Schubert variety in an orthogonal Grassmannian, we compute the multiplicity, more generally the Hilbert function. We first translate the problem from geometry to combinatorics by applying standard monomial theory. The solution of the resulting combinatorial problem forms the bulk of the paper. This approach has been followed earlier to solve the same problem for Grassmannians and symplectic Grassmannians. As an application, we present an interpretation of the multiplicity as the number of non-intersecting lattice paths of a certain kind. A more important application, although it does not appear here but elsewhere, is to the computation of the initial ideal, with respect to certain convenient monomial orders, of the ideal of the tangent cone to the Schubert variety.
Keywords: keywords orthogonal Grassmannian, Schubert variety, Hilbert function, multiplicity, Pfaffian ideal, O-domination, O-depth 
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     title = {Hilbert functions of points on {Schubert} varieties in orthogonal grassmannians},
     journal = {Journal of Algebraic Combinatorics},
     pages = {355--409},
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Raghavan, K.N.; Upadhyay, Shyamashree. Hilbert functions of points on Schubert varieties in orthogonal grassmannians. Journal of Algebraic Combinatorics, Tome 31 (2010) no. 3, pp. 355-409. http://geodesic.mathdoc.fr/item/JAC_2010__31_3_a2/