Sym-Noetherianity for powers of GL-varieties

TL;DR

This paper proves that certain infinite-dimensional spaces with combined symmetric and linear group actions are topologically Noetherian, extending finiteness results in algebraic geometry.

Contribution

It introduces a unified framework for studying spaces acted upon by both symmetric and general linear groups and proves their topological Noetherianity.

Findings

Spaces are topologically Noetherian under combined group actions

Extends finiteness properties to more general infinite-dimensional varieties

Provides a new approach to symmetry and algebraic geometry interactions

Abstract

Much recent literature concerns finiteness properties of infinite-dimensional algebraic varieties equipped with an action of the infinite symmetric group, or of the infinite general linear group. In this paper, we study a common generalisation in which the product of both groups acts on infinite-dimensional spaces, and we show that these spaces are topologically Noetherian with respect to this action.