Resolutions of orthogonal and symplectic analogues of determinantal ideals

We consider orthogonal and symplectic analogues of determinantal varieties . Such varieties simultaneously generalize usual determinantal varieties and rank varieties of symmetric or anti-symmetric matrices. We find (non-minimal) resolutions of the coordinate rings of the varieties . We determine that “nearly all” such varieties are Cohen–Macaulay and for those that are Cohen–Macaulay we calculate the type. Furthermore, we provide a simple characterization for which varieties are Gorenstein. As an application, we present a class of ideals in k[Hom(E,F)] that are Gorenstein of codimension 4.