Initial ideals of tangent cones to Richardson varieties in the Orthogonal Grassmannian via a Orthogonal-Bounded-RSK-Correspondence

A Richardson variety $X_\ga^\gc$ in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety $X^\gc$ in the Orthogonal Grassmannian and a opposite Schubert variety $X_\ga$ therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any $T$-fixed point of $X_\ga^\gc$, thus generalizing a result of Raghavan-Upadhyay \cite{Ra-Up2}. Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal bounded RSK (OBRSK). The OBRSK correspondence will give a degree-preserving bijection between a set of monomials defined by the initial ideal of the ideal of the tangent cone (as mentioned above) and a `standard monomial basis'. A similar work for Richardson varieties in the ordinary Grassmannian was done by Kreiman in \cite{Kr-bkrs}.