Abstract
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).
Highlights
A Richardson variety Xαγ in the Orthogonal Grassmannian
We provide an explicit description of the initial ideal for the ideal of the tangent cone at any T-fixed point eβ of Xαγ
It should be noted that the local properties of the Schubert varieties at T-fixed points determine their local properties at all other points, because of the B action; but this does not extend to the Richardson varieties, since Richardson varieties only have a T-action
Summary
Sturmfels [4] and Herzog and Trung [5] proved results on a class of determinantal varieties which are equivalent to the results of [6,7,8] for the case of the Schubert varieties (in the ordinary Grassmannian) at the T-fixed point eid The key to their proofs was to use a version of the Robinson-SchenstedKnuth 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.”. Kreiman has mentioned in his paper [8] that he believes in the possibility of adapting the methods of his paper ([8]) to the Richardson varieties in the Symplectic and the Orthogonal Grassmannian as well; this paper supports Kreiman’s belief for the Orthogonal Grassmannian case
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