Quadratic stabilization of the canonical Foulkes–Howe map

For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map ${\mathrm{Sym}}^b({\mathrm{Sym}}^{a}V)⟶{\mathrm{Sym}}^a({\mathrm{Sym}}^{b}V)$ is surjective whenever a ≥ 2 and $b\ge a(a-1)$. This gives a quadratic stabilization bound independent of $\dim V$.