Result 210, Algebra

Foulkes' conjecture for sixth powers and quadratic stabilization

Proves the sixth case of Foulkes’ conjecture: Sym6(SymbV)\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^bV) embeds equivariantly in Symb(Sym6V)\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 Symb(SymaV)→Syma(SymbV)\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≥a(a−1)b\ge a(a-1), giving dimension-independent quadratic stabilization.

Lean formalization Proof

The bigger picture

Why it matters

Symmetric powers describe combinations of vectors in which order does not matter. These manuscripts compare two ways of nesting that construction, asking whether every symmetry pattern in one also appears in the other.

What changes?

The unreviewed manuscripts report that, for every finite-dimensional complex vector space and integer b at least 6, the sixth symmetric power of the bth embeds into the reverse order, respecting invertible linear coordinate changes. They also claim the canonical multiplication map from the bth symmetric power of the ath to the reverse order is onto whenever the integers satisfy a at least 2 and b at least a times (a minus 1). This bound is independent of dimension.

What does that help mathematicians do?

For a equal to 6, the general bound guarantees surjectivity only from b equal to 30 onward. The separate sixth-power result supplies embeddings also for b from 6 through 29, without claiming that the canonical map is onto there. An embedding means no irreducible symmetry type occurs more often in the source than in the target, giving researchers concrete constraints on how these nested constructions decompose.

Are there practical applications?

The immediate value is foundational: the claims clarify how polynomial constructions organize under changes of coordinates. For each fixed a, the quadratic bound confines possible failures of surjectivity to a finite range of exponents b, uniformly across dimensions. This focuses further investigation without settling Foulkes' conjecture in general.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Foulkes' conjecture for the sixth symmetric power

September 25, 2026 36 pages

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 GL(V)\mathop{\mathrm{GL}}\nolimits (V)-equivariant injection Sym6(SymbV)↪Symb(Sym6V)\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^b V)\hookrightarrow\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6 V).

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Quadratic stabilization of the canonical Foulkes--Howe map

September 25, 2026 7 pages Main result formalized in Lean

For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map Symb(SymaV)⟶Syma(SymbV)\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≥a(a−1)b\ge a(a-1). This gives a quadratic stabilization bound independent of dim⁡V\dim V.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/210.md.

Foulkes' conjecture for sixth powers and quadratic stabilization

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result proves surjectivity of the canonical averaged Foulkes–Howe map Symb(Syma V)→Syma(Symb V)\mathrm{Sym}^b(\mathrm{Sym}^a\,V)\to\mathrm{Sym}^a(\mathrm{Sym}^b\,V) for every finite-dimensional complex vector space, a≥2a\ge2, and b≥a(a−1)b\ge a(a-1). It also covers the bijective a=1a=1 case, the vanishing-on-products consequence, and the corresponding equivariant embedding in the reverse direction. The sixth-power specialization is covered for b≥30b\ge30; the companion's full range b≥6b\ge6 is not included.

Comparator links

Result Comparator statement
Quadratic stabilization of the canonical Foulkes–Howe map FoulkesHowe.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.