Vol. 12, No. 2, 2018

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

Nicolas Ford, Jake Levinson and Steven V Sam

Vol. 12 (2018), No. 2, 285–303
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 GLk-equivariant Betti tables of modules over the coordinate ring of k × n matrices, and, dually, cohomology tables of vector bundles on the Grassmannian Gr(k, 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–Fløystad–Weyman.

Keywords
Boij–Söderberg theory, Betti table, cohomology table, Schur functors, Grassmannian, free resolutions, equivariant K-theory
Mathematical Subject Classification 2010
Primary: 13D02
Secondary: 05E99
Milestones
Received: 21 August 2016
Revised: 4 December 2017
Accepted: 3 January 2018
Published: 13 May 2018
Authors
Nicolas Ford
Mathematics Department
University of California
Berkeley, CA
United States
Jake Levinson
Mathematics Department
University of Michigan
Ann Arbor, MI
United States
Mathematics Department
University of Washington
Seattle, WA
United States
Steven V Sam
Mathematics Department
University of Wisconsin
Madison, WI
United States