Topological noetherianity for powers of algebraic representations
Pith reviewed 2026-05-16 09:18 UTC · model grok-4.3
The pith
Powers of algebraic representations of the orthogonal and symplectic groups are topologically Noetherian under the action of the symmetric group times the group.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The central claim is that powers of a polynomial GL-representation are topologically Noetherian under Sym × GL, and this extends to powers of algebraic representations of the orthogonal and symplectic groups under Sym × O and Sym × Sp respectively. The work provides further evidence that infinite powers of topologically Noetherian varieties remain topologically Noetherian up to permutations.
What carries the argument
Topological Noetherianity of the representation variety, which requires every descending chain of closed subsets to stabilize, under the joint action of Sym and the group G.
Load-bearing premise
The topological Noetherianity results for the base cases in the cited prior works hold, and the representations considered are algebraic or polynomial as stated.
What would settle it
An explicit algebraic representation of O or Sp together with a descending chain of closed subsets in one of its powers that does not stabilize under Sym × O or Sym × Sp would falsify the claim.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that if V is an algebraic representation of the orthogonal group O or the symplectic group Sp, then the infinite power V^∞ is topologically Noetherian under the natural action of Sym × O (respectively Sym × Sp). The argument reduces the claim to the known base cases for GL via highest-weight theory, explicit stabilizer computations, and adaptation of the Sym-action and the relevant inverse-limit topology; it builds directly on the results of arXiv:2212.05790 and arXiv:1708.06420.
Significance. If the reduction steps hold, the result supplies further concrete evidence for the general principle that infinite powers of topologically Noetherian varieties remain topologically Noetherian after permutation of coordinates. The extension from GL to the other classical groups is a natural and useful increment; the paper correctly identifies that its soundness rests on the correctness of the two cited prior works, which is the expected architecture for such an extension.
minor comments (3)
- [§2.3] §2.3: the precise definition of the inverse-limit topology on the infinite power is referenced to the GL paper but not restated; a one-sentence recap would improve readability for readers who have not consulted the earlier work.
- [Proposition 4.2] Proposition 4.2: the stabilizer computation for the symplectic case is stated in terms of root-system data; an explicit matrix representative for a generic point in the orbit would make the subsequent Noetherianity argument easier to follow.
- [Bibliography] The abstract and introduction both cite arXiv:2212.05790 and arXiv:1708.06420; the bibliography entry for the second preprint should include its published version if it has appeared.
Simulated Author's Rebuttal
We thank the referee for the positive report and recommendation of minor revision. The assessment correctly situates the work as a natural extension of the GL case via highest-weight theory and stabilizer computations, building directly on the cited prior results. No specific major comments were raised.
Circularity Check
No significant circularity; extension relies on independent prior results
full rationale
The manuscript extends topological Noetherianity from the GL case in the cited arXiv:2212.05790 to algebraic representations of O and Sp via standard highest-weight theory, stabilizer computations, and adaptation of the Sym-action and topology. No equations or steps reduce by construction to fitted parameters, self-definitions, or unverified self-citations within this paper. The derivation chain is self-contained against the external base cases, which are treated as given independent inputs rather than derived here.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Topological Noetherianity holds for the base polynomial representations of GL, O, and Sp as established in the cited prior works.
discussion (0)
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