On the geometry and representation theory of isomeric matrices

TL;DR

This paper explores the geometric and representation-theoretic properties of isomeric matrices, extending classical results to the superalgebra setting with connections to the queer supergroup and twisted commutative algebras.

Contribution

It establishes analogs of classification and noetherian properties for equivariant modules of isomeric matrices under the queer supergroup, generalizing prior work on ordinary matrices.

Findings

Classification of equivariant ideals for isomeric matrices

Noetherian property for families of equivariant modules

Connections to Brauer category and twisted commutative algebras

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.