Syzygies of the apolar ideals of the determinant and permanent

TL;DR

This paper studies the syzygies of the apolar ideals of the determinant and permanent polynomials, revealing their generators, relations, and Betti numbers, with implications for algebraic complexity.

Contribution

It extends previous work by characterizing the minimal generating sets and relations of these ideals, and provides explicit formulas and conjectures for their Betti numbers.

Findings

Linear relations generate the syzygies of the determinant apolar ideal.

A minimal generating set of linear and quadratic relations is provided for the permanent apolar ideal.

Explicit formulas for Betti numbers are given, along with conjectural descriptions.

Abstract

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.