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

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

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.