Hilbert functions of points on Schubert varieties in orthogonal Grassmannians

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. Taking the Schubert variety to be of a special kind and the point to be the ‘identity coset,’ our problem specializes to one about Pfaffian ideals, treatments of which by different methods exist in the literature. Also available in the literature is a geometric solution when the point is a ‘generic singularity.’