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.