Towards Boij–Söderberg theory for Grassmannians: the case of square matrices

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–Söderberg theory for Grassmannians,” with the goal of characterizing the cones of ${GL}_k$-equivariant Betti tables of modules over the coordinate ring of $k\times n$ matrices, and, dually, cohomology tables of vector bundles on the Grassmannian $Gr\left(k,{ℂ}^n\right)$. 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–Fløystad–Weyman.