Syzygies of the apolar ideals of the determinant and permanent

Abstract

We investigate the space of syzygies of the apolar ideals ${\det }_n^{\perp }$ and ${\mathrm{perm}}_n^{\perp }$ of the determinant ${\det }_n$ and permanent ${\mathrm{perm}}_n$ 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 ${\mathrm{perm}}_n^{\perp }$ do not generate all relations, but we provide a minimal generating set of linear and quadratic relations. For both ${\det }_n^{\perp }$ and ${\mathrm{perm}}_n^{\perp }$, 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.

29 pages, comments welcome