Horn conditions for quiver subrepresentations and the moment map

Michael Walter (Faculty of Computer Science, Ruhr University, Bochum, Germany; and Korteweg-de Vries Institute for Mathematics, Institute for Theoretical Physics, Institute for Logic, Language & Computation & QuSoft, University of Amsterdam, The Netherlands)

Abstract

We give inductive conditions that characterize the Schubert positions of subrepresentations of a general quiver representation. Our results generalize Belkale’s criterion for the intersection of Schubert varieties in Grassmannians and refine Schofield’s characterization of the dimension vectors of general subrepresentations. This implies Horn type inequalities for the moment cone associated to the linear representation of the group $G = \prod_x \mathrm{GL}(n_x)$ associated to a quiver and a dimension vector $n = (n_x)$.

2010 Mathematics Subject Classification

Primary 14N15, 53D20. Secondary 15A42, 16G20, 22E47.

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We are delighted to include this article as a tribute to the always inspiring work of Victor Guillemin

V. Baldoni acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006 and the partial support of a PRIN2015 grant. M. Walter acknowledges support by the NWO through Veni grant 680-47-459 and grant OCENW.KLEIN.267, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972, and by the European Research Council (ERC) through ERC Starting Grant 101040907-SYMOPTIC.

Received 20 July 2021

Accepted 11 August 2022

Published 20 November 2023