Towards Boij-S\"oderberg theory for Grassmannians: the case of square matrices

TL;DR

This paper characterizes the cone of GL-equivariant Betti tables for Cohen-Macaulay modules over square matrix coordinate rings, laying groundwork for a broader Boij-S"oderberg theory for Grassmannians.

Contribution

It provides the first characterization of extremal rays of Betti table cones for square matrices, using combinatorial and geometric methods, as a base case for a general theory.

Findings

Identified extremal rays using Hall's Theorem

Constructed equivariant free resolutions via Weyman's technique

Established foundational results for Grassmannian Boij-S"oderberg theory

Abstract

We characterize the cone of GL-equivariant Betti tables of Cohen-Macaulay modules of codimension 1, up to rational multiple, over the coordinate ring of square matrices. This result serves as the base case for `Boij-S\"oderberg theory for Grassmannians', with the goal of characterizing the cones of GL_k-equivariant Betti tables of modules over the coordinate ring of k x n matrices, and, dually, cohomology tables of vector bundles on the Grassmannian Gr(k, C^n). The proof uses Hall's Theorem on perfect matchings in bipartite graphs to compute the extremal rays of the cone, and constructs the corresponding equivariant free resolutions by applying Weyman's geometric technique to certain graded pure complexes of Eisenbud-Fl{\o}ystad-Weyman.