On the geometry and representation theory of isomeric matrices

Abstract

The space of $n \times m$ complex matrices can be regarded as an algebraic variety on which the group ${\bf GL}_n \times {\bf GL}_m$ 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. In this paper, we establish analogs of these results for the space of $(n|n) \times (m|m)$ isomeric matrices with respect to the action of ${\bf Q}_n \times {\bf Q}_m$, where ${\bf Q}_n$ 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.