Syzygies of the apolar ideals of the determinant and permanent

We investigate the space of syzygies of the apolar ideals $\det_n^\perp$ and ${\rm perm}_n^\perp$ of the determinant $\det_n$ and permanent ${\rm perm}_n$ 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 $\det_n^{\perp}$ is generated by linear relations and we describe a minimal generating set. The linear relations of ${\rm perm}_n^{\perp}$ do not generate all relations, but we provide a minimal generating set of linear and quadratic relations. For both $\det_n^\perp$ and ${\rm perm}_n^\perp$, we give formulas for the Betti numbers $\beta_{1,j}$, $\beta_{2,j}$ and $\beta_{3,4}$ for all $j$ as well as conjectural descriptions of other Betti numbers. Finally, we provide representation-theoretic descriptions of certain spaces of linear syzygies.