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

Ford, Nicolas; Levinson, Jake; Sam, Steven

doi:10.2140/ant.2018.12.285

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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

DOI: 10.2140/ant.2018.12.285

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 $Gr (k, ℂ^{n})$ . 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–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

Vol. 12, No. 2, 2018