Foundations of Boij-S\"oderberg Theory for Grassmannians

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.