Vol. 16, No. 6, 2022

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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
Abstract

The space of n × m complex matrices can be regarded as an algebraic variety on which the group GL n × 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) × (m|m) isomeric matrices with respect to the action of Qn ×Qm, where Qn 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