OpenAI · 210-01 · 2026-09-25 · Algebra

Foulkes' conjecture for the sixth symmetric power

preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026

Family 210. Foulkes’ conjecture for sixth powers and quadratic stabilization.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves the sixth case of Foulkes’ conjecture: $Sym^{6} (Sym^{b} V)$ embeds equivariantly in $Sym^{b} (Sym^{6} V)$ for every $b \geq 6$ and finite-dimensional complex $V$ . More generally, the canonical multiplication map $Sym^{b} (Sym^{a} V) \to Sym^{a} (Sym^{b} V)$ is surjective for $a \geq 2$ and $b \geq a (a - 1)$ , giving dimension-independent quadratic stabilization.

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 $GL (V)$ -equivariant injection $Sym^{6} (Sym^{b} V) ↪ Sym^{b} (Sym^{6} V)$ .

Preprint source and Lean scope

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A Lean scope note is stored for family 210, but it does not name this preprint. It names 210-02. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.