Foundations of Boij-S\"oderberg Theory for Grassmannians

TL;DR

This paper extends Boij-S"oderberg theory to $GL_k$-equivariant modules on Grassmannians, introducing new algebraic tools and pairings, and explores extremal rays for specific matrix sizes.

Contribution

It develops equivariant analogues of key Boij-S"oderberg features and generalizes previous results to complex modules beyond free resolutions.

Findings

Extended Herzog-K"uhl equations to equivariant setting

Established a new proof of the pairing using graph matchings

Identified extremal rays for $2 imes 3$ matrices

Abstract

Boij-S\"oderberg theory characterizes syzygies of graded modules and sheaves on projective space. This paper continues earlier work with S. Sam, extending the theory to the setting of $GL_k$-equivariant modules and sheaves on Grassmannians. Algebraically, we study modules over a polynomial ring in $k n$ variables, thought of as the entries of a $k \times n$ matrix. We give equivariant analogues of two important features of the ordinary theory: the Herzog-K\"uhl equations and the pairing between Betti and cohomology tables. As a necessary step, we also extend previous results, concerning the base case of square matrices, to cover complexes other than free resolutions. Our statements specialize to those of ordinary Boij-S\"oderberg theory when $k=1$. Our proof of the equivariant pairing gives a new proof in the graded setting: it relies on finding perfect matchings on certain graphs associated to Betti tables. Finally, we give preliminary results on $2 \times 3$ matrices, exhibiting certain classes of extremal rays on the cone of Betti tables.