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Noetherianity for powers of algebraic representations

Alessandro Danelon

TL;DR

This work establishes topological Noetherianity for powers of algebraic representations under the action of the infinite symmetric group together with classical groups. By first linking Noetherianity across $\operatorname{GL}$, $\operatorname{O}$, and $\operatorname{Sp}$ via a group-homomorphism framework, the authors reduce the problem to proving the GL case, then extend the argument to the orthogonal and symplectic settings. The core method combines orbit-closure techniques and a $Z^+$-construction to transfer Noetherianity from finite-level truncations $(M\oplus E)^{\mathbb{N}}$ to the infinite-dimensional setting, building on and generalizing prior work of Draisma, Eggermont–Snowden, and Chiu–Danelon–Draisma–Eggermont–Farooq. The results broaden the class of tensor varieties known to be topologically Noetherian under $\operatorname{Sym} \times G$ actions and connect to the broader program on Noetherianity in infinite-dimensional representation theory. This has potential implications for understanding stability phenomena in tensor categories and invariant theory for infinite-dimensional group actions.

Abstract

Powers of a polynomial $\operatorname{GL}$-representation are topologically Noetherian under the action of $\operatorname{Sym} \times \operatorname{GL}$. We show that this result extends to powers of algebraic representations of the orthogonal and the symplectic groups. This work is a natural follow-up to arXiv:2212.05790 and to arXiv:1708.06420.

Noetherianity for powers of algebraic representations

TL;DR

This work establishes topological Noetherianity for powers of algebraic representations under the action of the infinite symmetric group together with classical groups. By first linking Noetherianity across , , and via a group-homomorphism framework, the authors reduce the problem to proving the GL case, then extend the argument to the orthogonal and symplectic settings. The core method combines orbit-closure techniques and a -construction to transfer Noetherianity from finite-level truncations to the infinite-dimensional setting, building on and generalizing prior work of Draisma, Eggermont–Snowden, and Chiu–Danelon–Draisma–Eggermont–Farooq. The results broaden the class of tensor varieties known to be topologically Noetherian under actions and connect to the broader program on Noetherianity in infinite-dimensional representation theory. This has potential implications for understanding stability phenomena in tensor categories and invariant theory for infinite-dimensional group actions.

Abstract

Powers of a polynomial -representation are topologically Noetherian under the action of . We show that this result extends to powers of algebraic representations of the orthogonal and the symplectic groups. This work is a natural follow-up to arXiv:2212.05790 and to arXiv:1708.06420.
Paper Structure (7 sections, 8 theorems, 3 equations)

This paper contains 7 sections, 8 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Description of the spaces
  4. Connection to the literature
  5. The proof strategy
  6. Auxiliary notation and results
  7. The general linear case

Key Result

Theorem 1.1

Let $G$ be one of the groups $\operatorname{GL}, \operatorname{O},$ or $\operatorname{Sp}$. Let $E$ be an algebraic $G$-representation. Consider the action of the group $\operatorname{Sym} \times G$ on the countable direct sum $E^{\oplus \mathbb{N}}$ with $\operatorname{Sym}$ permuting the copies o

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2: chiu-danelon-draisma-eggermont-farooq
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • proof
  • Lemma 1.6
  • proof
  • ...and 4 more