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Landsberg's Problem 7.19: a polynomial stabilization bound for the canonical Foulkes-Howe map

For a,b≥1a,b\ge1 and a finite-dimensional complex vector space VV, the canonical Foulkes-Howe (Hadamard-Howe) map μa,b,V:Symb(SymaV)→Syma(SymbV)\mu_{a,b,V}:\mathrm{Sym}^b(\mathrm{Sym}^aV)\to\mathrm{Sym}^a(\mathrm{Sym}^bV) is induced by multiplication in (SymV)⊗a(\mathrm{Sym}V)^{\otimes a}; its image is the degree-bb part of the coordinate ring of the Chow variety of products of aa linear forms, whose normalization is the whole target. Surjectivity in degree bb gives an equivariant embedding as in Foulkes' conjecture. Brion proved eventual surjectivity and later an effective bound depending on both aa and dim⁡V\dim V. Landsberg's survey of geometric complexity theory (Problem 7.19) asks for a polynomial bound. Is there a degree bound, polynomial in aa and valid in every dimension, beyond which μa,b,V\mu_{a,b,V} is surjective?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Representation theory; Chow varieties and geometric complexity theory
Posed by
J. M. Landsberg (Geometric complexity theory: an introduction for geometers, Ann. Univ. Ferrara 2015, Problem 7.19)
Year posed
2015
Years open
11y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
17 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1: for every a≥2a\ge2, b≥a(a−1)b\ge a(a-1) and finite-dimensional complex VV, μa,b,V\mu_{a,b,V} is surjective; equivalently the Chow-variety coordinate ring agrees with its normalization in degree bb. In Landsberg's notation the bound is d≥n(n−1)d\ge n(n-1), in every dimension. By complete reducibility this yields Foulkes' embedding Syma(SymbV)↪Symb(SymaV)\mathrm{Sym}^a(\mathrm{Sym}^bV)\hookrightarrow\mathrm{Sym}^b(\mathrm{Sym}^aV) for b≥a(a−1)b\ge a(a-1). The least stabilization degree is not determined, the bound is not claimed sharp, and nothing is said about injectivity of the map in the other direction (which fails at (5,5)(5,5) and (6,6)(6,6)).

What the AI did

The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The stabilization theorem is also the large-degree input to the companion proving the sixth case of Foulkes' conjecture (separate entry). The release includes a Lean formalization of it.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1 and the history paragraph were read against Landsberg's request for a polynomial bound as the manuscript cites it. Lean: formalization.yaml lists ComparatorChallenges/FoulkesHowe (declaration OAI.Problem346.canonical_foulkes_howe_surjective, file OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean). The statement was read: for every finite-dimensional complex VV and a≥2a\ge2, b≥a(a−1)b\ge a(a-1), there is a unique linear map given by the averaged transpose formula on products, and it is surjective. That is the headline. Not rebuilt here. The paper does not claim the bound is sharp.

Sources

Changelog1 change

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