On the geometry and representation theory of isomeric matrices

Nagpal, Rohit; Sam, Steven V.; Snowden, Andrew

doi:10.2140/ant.2022.16.1501

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On the geometry and representation theory of isomeric matrices

Rohit Nagpal, Steven V. Sam and Andrew Snowden

Vol. 16 (2022), No. 6, 1501–1529

DOI: 10.2140/ant.2022.16.1501

Abstract

The space of $n \times m$ complex matrices can be regarded as an algebraic variety on which the group ${GL}_{n} \times {GL}_{m}$ acts. There is a rich interaction between geometry and representation theory in this example. In an important paper, de Concini, Eisenbud, and Procesi classified the equivariant ideals in the coordinate ring. More recently, we proved a noetherian result for families of equivariant modules as $n$ and $m$ vary. We establish analogs of these results for the space of $(n | n) \times (m | m)$ isomeric matrices with respect to the action of $Q_{n} \times Q_{m}$ , where $Q_{n}$ is the automorphism group of the isomeric structure (commonly known as the “queer supergroup”). Our work is motivated by connections to the Brauer category and the theory of twisted commutative algebras.

Keywords

Lie superalgebras, twisted commutative algebras, isomeric algebra

Mathematical Subject Classification

Primary: 13A50, 13E05

Milestones

Received: 2 February 2021

Revised: 21 August 2021

Accepted: 18 September 2021

Published: 27 September 2022

Authors

Rohit Nagpal
Department of Mathematics University of Michigan Ann Arbor, MI United States

Steven V. Sam
Department of Mathematics University of California San Diego, CA United States

Andrew Snowden
Department of Mathematics University of Michigan Ann Arbor, MI United States

Vol. 16, No. 6, 2022