Initial Ideals of Tangent Cones to the Richardson Varieties in the Orthogonal Grassmannian

A Richardson variety in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety in the Orthogonal Grassmannian and an opposite Schubert variety 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 , thus generalizing a result of Raghavan and Upadhyay (2009). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal-bounded-RSK (OBRSK). 1. Introduction The Orthogonal Grassmannian is defined in Section 2. A Richardson variety in the Orthogonal Grassmannian (The Richardson varieties in the ordinary Grassmannian are also studied by Stanley in [1], where these varieties are called skew Schubert varieties. Discussion of these varieties in the ordinary Grassmannian also appears in [2].) is defined to be the intersection of a Schubert variety in the Orthogonal Grassmannian with an opposite Schubert variety therein. In particular, the Schubert and opposite Schubert varieties are special cases of the Richardson varieties. In this paper, we provide an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any -fixed point of . It should be noted that the local properties of the Schubert varieties at -fixed points determine their local properties at all other points, because of the action; but this does not extend to the Richardson varieties, since Richardson varieties only have a -action. In Raghavan and Upadhyay [3], an explicit description of the initial ideal (with respect to certain conveniently chosen term orders) for the ideal of the tangent cone at any -fixed point of a Schubert variety in the Orthogonal Grassmannian has been obtained. In this paper, we generalize the result of [3] to the case of the Richardson varieties in the Orthogonal Grassmannian. Sturmfels [4] and Herzog and Trung [5] proved results on a class of determinantal varieties which are equivalent to the results of [6–8] for the case of the Schubert varieties (in the ordinary Grassmannian) at the -fixed point . The key to their proofs was to use a version of the Robinson-Schensted-Knuth correspondence (which we will call the “ordinary” RSK) in order to establish a degree-preserving bijection between a set of monomials defined by an initial ideal and a “standard monomial basis.” The difficulty in extending this method of proof to the case of the Schubert varieties (in the ordinary Grassmannian) at an