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Foulkes' conjecture for the sixth symmetric power
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Abstract
We prove the sixth-symmetric-power case of Foulkes' conjecture. For every integer b ≥ 6 and every finite-dimensional complex vector space V, there is a $\mathop{\mathrm{GL}}\nolimits (V)$-equivariant injection $\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^b V)\hookrightarrow\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6 V)$.
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Notes
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Automatically collected notes (unreviewed)
work in progress
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.
Follow-up works
- realopjt, Independent check of the base range in "Foulkes' Conjecture for the Sixth Symmetric Power" (GitHub, 2026). This Claude-assisted Rust checker reports exact weight-basis computations confirming the multiplicity inequalities for every $6\le b\le25$, while explicitly excluding independent verification of the higher-range certificates and general argument.
Formalization
- OpenAI, Foulkes' conjecture for sixth powers and quadratic stabilization: Lean scope notes (OpenAI math release, lean/docs/210.md, 2026-10-06). The scope note describes formalized canonical surjectivity for $a\ge2$ and $b\ge a(a-1)$ and the reverse equivariant embedding, covering the sixth-power case for $b\ge30$; per the notes, "the companion's full range $b\ge6$ is not included". [family-level]
Possibly relevant literature (found by the agent, not checked)
- C. Bowman, R. Paget, The partition algebra and the plethysm coefficients I: stability and Foulkes' conjecture (arXiv:1809.08128, 2018, rev. 2022). Proves a strengthened Foulkes' conjecture for stable plethysm coefficients, which bears on the stable range that the paper treats via Brion's eventual-truth result and the companion's stabilization bound.
- 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 the coordinate ring of a normalization, both relevant to the paper's discussion of the canonical map and of Chow-variety normalization.
- A. Abdesselam, J. Chipalkatti, Brill–Gordan loci, transvectants and an analogue of the Foulkes conjecture (arXiv:math/0411110, 2004). Proves surjectivity of an analogue of the Foulkes–Howe map for binary forms; the paper's cited reference RaicuSamWeymanYang2026 generalizes this work, but the paper does not cite it directly.
- S. Sarkar, Stability of plethysm coefficients and modified polynomial induction (arXiv:2610.07929, October 2026). Gives sharper stabilization bounds for coefficients of $h_n[h_m]$ and a short proof of Brion's stability theorem, which the paper cites for eventual truth at fixed a; it does not mention the release.
News and blogs
- realopjt, Checking an AI's maths, part 2 (Checking AI maths, 2026-10-07). This post, prepared with Claude's help, explains the independent check of $6\le b\le25$ and reports a successful rerun of the manuscript's programs, distinguishing those checks from verification of the remaining proof.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 210
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026,
author = {{OpenAI}},
title = {{Foulkes' conjecture for the sixth symmetric power}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf}{OAI:Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026}},
year = {2026}
}From OpenAI's README for this paper:
Verification
Requires Python 3.9 or later. Run the supplied read-only checks from this paper directory:
python3 -B verification/verify_computations.py --check
Success prints PASS and confirms the bundled source and reference-stream
identities, all 62 expected numeric rows, agreement with the manuscript tables,
and the stated numerical bounds. This command does not execute the C++ programs.
To compile and run both complete exact computations, GNU C++ with signed 128-bit integers and Boost headers is also required:
python3 -B verification/verify_computations.py --run --serial
Alternatively, replace --serial with --threads 2 to use OpenMP for Appendix A.
Assertions remain enabled. A successful full run prints PASS after both
programs exit successfully and their complete output streams match the reference
bytes. It records compiler and execution details in a new temporary directory
outside the paper; --output PATH selects a new external directory instead.
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:
- Quadratic stabilization of the canonical Foulkes--Howe map · September 25, 2026