Equivariant prime ideals for infinite dimensional supergroups

TL;DR

This paper develops a framework for understanding equivariant prime ideals in infinite dimensional supergroup actions, revealing new structural insights and applications to queer determinantal ideals.

Contribution

It generalizes previous results to a broader class of supergroups, providing a unified approach to equivariant prime ideals in infinite dimensional settings.

Findings

Pathologies in G-primes can be resolved in super vector spaces.

Framework applies to various supergroups beyond GL.

Results have applications to queer determinantal ideals.

Abstract

Let $A$ be a commutative algebra equipped with an action of a group $G$. The so-called $G$-primes of $A$ are the equivariant analogs of prime ideals, and of central importance in equivariant commutative algebra. When $G$ is an infinite dimensional group, these ideals can be very subtle: for instance, distinct $G$-primes can have the same radical. In previous work, the second author showed that if $G$ is ${\bf GL}$ and $A$ is a polynomial representation, then these pathologies disappear when working with super vector spaces; this leads to a geometric description of $G$-primes of $A$. In the present paper, we construct an abstract framework around this result, and apply the framework to prove analogous results for other (super)groups. We give some applications to the isomeric determinantal ideals (more commonly known as "queer determinantal ideals").