Let N be a positive integer, and let m,n be integers with m leqslant n. For representations of operatornameSL_N(C), write U leqslant V when U is isomorphic to a subrepresentation of V. Foulkes's conjecture. operatornameSym^m operatornameSym^n C^N leqslant operatornameSym^n operatornameSym^m C^N. This is a central special case of the plethysm problem and is stated in the source as open. It concerns comparing the decompositions of two iterated symmetric powers as operatornameSL_N(C)-representations.

Foulkes's conjecture for symmetric powers

About 1 year old · traced to 2025

Let $N$ be a positive integer, and let $m,n$ be integers with $m\leqslant n$ . For representations of $\operatorname{SL}_N(\mathbb{C})$ , write $U\leqslant V$ when $U$ is isomorphic to a subrepresentation of $V$ . Foulkes's conjecture.

\operatorname{Sym}^m \operatorname{Sym}^n \mathbb{C}^N\leqslant \operatorname{Sym}^n \operatorname{Sym}^m \mathbb{C}^N.

This is a central special case of the plethysm problem and is stated in the source as open. It concerns comparing the decompositions of two iterated symmetric powers as $\operatorname{SL}_N(\mathbb{C})$ -representations.

References

Primary source

Moritz Gangl, Álvaro Gutiérrez and Michał Szwej, “On the generalised Foulkes conjecture for SL_2(C) under divisibility conditions”, arXiv:2507.06220 (2026).

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RemarkAI-assistedClaimed by OpenAI. For every finite-dimensional complex vector space V and every b>=6, the manuscript claims a GL(V)-equivariant injection Sym^6(Sym^b(V)) into Sym^b(Sym^6(V)). Restricting to SL_N(C) gives the page’s Foulkes inequality for m=6 and n=b. The manuscript addresses this fixed inner symmetric power; it does not establish every m<=n.See full solutionHide full solution

Claimed by OpenAI. For every finite-dimensional complex vector space V and every b>=6, the manuscript claims a GL(V)-equivariant injection Sym^6(Sym^b(V)) into Sym^b(Sym^6(V)). Restricting to SL_N(C) gives the page’s Foulkes inequality for m=6 and n=b. The manuscript addresses this fixed inner symmetric power; it does not establish every m<=n.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf

OpenAI-210-01-Foulkes-conjecture-for-the-sixth-symmetric-power.pdf451,463 bytesOpen

RemarkAI-assistedClaimed by OpenAI. For complex V, the manuscript claims the canonical Foulkes–Howe map Sym^b(Sym^a(V)) to Sym^a(Sym^b(V)) is surjective for a>=2 and b>=a(a-1). Semisimplicity then gives the page’s opposite-direction representation inclusion for that range. This is quadratic stabilization; it does not cover every a<=b or assert that the canonical map in the opposite direction is injective.See full solutionHide full solution

Claimed by OpenAI. For complex V, the manuscript claims the canonical Foulkes–Howe map Sym^b(Sym^a(V)) to Sym^a(Sym^b(V)) is surjective for a>=2 and b>=a(a-1). Semisimplicity then gives the page’s opposite-direction representation inclusion for that range. This is quadratic stabilization; it does not cover every a<=b or assert that the canonical map in the opposite direction is injective.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf

OpenAI-210-02-Quadratic-stabilization-of-the-canonical-Foulkes-Howe-map.pdf238,511 bytesOpen