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Quadratic stabilization of the canonical Foulkes--Howe map
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Abstract
For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map $\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^a V)\longrightarrow\mathop{\mathrm{Sym}}\nolimits ^a(\mathop{\mathrm{Sym}}\nolimits ^b V)$ is surjective whenever a ≥ 2 and $b\ge a(a-1)$. This gives a quadratic stabilization bound independent of $\dim V$.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Foulkes' conjecture for sixth powers and quadratic stabilization: Lean scope notes (OpenAI math release, lean/docs/210.md, 2026-10-06). According to the scope notes, the release's Lean development formalizes this paper's main theorem, surjectivity of the canonical averaged Foulkes–Howe map for all finite-dimensional complex $V$, $a\ge2$, $b\ge a(a-1)$, together with the $a=1$ case, the vanishing-on-products consequence and the reverse equivariant embedding. [family-level]
Possibly relevant literature (found by the agent, not checked)
- H. Kadish, J. M. Landsberg, Padded polynomials, their cousins, and geometric complexity theory (arXiv:1204.4693, 2012). Proves asymptotic injectivity of the Foulkes–Howe map and describes explicitly the coordinate ring of a normalization, relevant to the paper's normalization interpretation ($A$ versus $R$) of the stabilization question.
- A. Abdesselam, J. Chipalkatti, Brill–Gordan loci, transvectants and an analogue of the Foulkes conjecture (arXiv:math/0411110, 2004). Proves surjectivity of an equivariant map between plethystic compositions of symmetric powers that is analogous to the Foulkes–Howe map, in the binary case.
- C. Raicu, S. V. Sam, J. Weyman, F. Yang, Powers of binary forms and derived Hermite reciprocity (arXiv:2602.15175, 2026). Generalizes Abdesselam–Chipalkatti's results on an analogue of the Foulkes–Howe map for binary forms; the companion paper cites it, but this paper does not.
Digests and expositions
- Math Referee, Quadratic stabilization of the canonical Foulkes–Howe map (reading guide, review dated 2026-10-13). This AI-assisted guide maps the annihilator criterion, differential injectivity lemma, two derivative sequences and equivariant splitting, with independent human assessment explicitly pending. [AI-generated per source]
- Perplexity / Orply, A Quadratic Bound Guarantees Surjectivity of the Foulkes–Howe Map (Orply, 2026-10-11). This AI-assisted exposition explains the derivative norm identity and degree bookkeeping behind the sufficient bound $b\ge a(a-1)$, including its specialization to $b\ge30$ when $a=6$. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 210
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026,
author = {{OpenAI}},
title = {{Quadratic stabilization of the canonical Foulkes--Howe map}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf}{OAI:Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/210.md. The Lean statements have not been checked by people against the paper's claims.
Family 210: Foulkes' conjecture for sixth powers and quadratic stabilization
Proves the sixth case of Foulkes’ conjecture: $\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^bV)$ embeds equivariantly in $\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6V)$ for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map $\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^aV)\to\mathop{\mathrm{Sym}}\nolimits ^a(\mathop{\mathrm{Sym}}\nolimits ^bV)$ is surjective for a ≥ 2 and $b\ge a(a-1)$, giving dimension-independent quadratic stabilization.
Other papers in this family:
- Foulkes' conjecture for the sixth symmetric power · September 25, 2026