Topological noetherianity for powers of algebraic representations

TL;DR

This paper extends the topological Noetherianity property of powers of algebraic representations from general linear groups to orthogonal and symplectic groups, supporting the idea that infinite powers preserve this property.

Contribution

It generalizes topological Noetherianity results to powers of algebraic representations of orthogonal and symplectic groups, beyond the previously known linear case.

Findings

Powers of algebraic representations of O and Sp groups are topologically Noetherian.

Supports the conjecture that infinite powers of topologically Noetherian varieties remain so.

Builds on prior work to broaden the class of groups for which topological Noetherianity holds.

Abstract

Powers of a polynomial $\operatorname{GL}$-representation are topologically Noetherian under the action of $\operatorname{Sym} \times \operatorname{GL}$. We extend this result to powers of algebraic representations of the orthogonal and the symplectic groups, proving topological Noetherianity under the action of $\operatorname{Sym} \times \operatorname{O}$ and $\operatorname{Sym} \times \operatorname{Sp}$ respectively. This work builds on arXiv:2212.05790 and arXiv:1708.06420, and it provides further evidence that infinite powers of topologically Noetherian varieties remain topologically Noetherian up to permutations.