Orthogonal and symplectic analogues of determinantal ideals

We consider subvarieties of determinantal varieties determined by an additional rank equation that defines an orthogonal or symplectic structure. Such varieties simultaneously generalize usual determinantal varieties and rank varieties of symmetric or anti-symmetric matrices. In this article, we find a non-trivial class of such orthogonal or symplectic analogues of determinantal varieties for which we can provide a completely combinatorial description of the terms in a minimal resolution of the coordinate ring. The results come as an application of the geometric technique and Bottʹs theorem for the cohomology of vector bundles over the Grassmannian.