Syzygies of the apolar ideals of the determinant and permanent

Abstract

We investigate the space of syzygies of the apolar ideals $${\text {det}}_n^\perp $$ and $${\mathrm{perm}}_n^\perp $$ of the determinant $${\text {det}}_n$$ 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 $${\text {det}}_n^{\perp }$$ 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 $${\text {det}}_n^\perp $$ 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.