Syzygies of the apolar ideals of the determinant and permanent

We investigate the space of syzygies of the apolar ideals detn perpendicular to and permn perpendicular to of the determinant detn and permanent permn polynomials. Shafiei had proved that these ideals are generated by quadrics and provided a minimal generating set. Extending on her work, in characteristic distinct from two, we prove that the space of relations of detn perpendicular to is generated by linear relations and we describe a minimal generating set. The linear relations of permn perpendicular to do not generate all relations, but we provide a minimal generating set of linear and quadratic relations. For both detn perpendicular to and permn perpendicular to, we give formulas for the Betti numbers beta 1,jfor all j as well as conjectural descriptions of other Betti numbers. Finally, we provide representation-theoretic descriptions of certain spaces of linear syzygies.