Foulkes' conjecture for the sixth symmetric power: $\mathrm{Sym}^6(\mathrm{Sym}^bV)\hookrightarrow\mathrm{Sym}^b(\mathrm{Sym}^6V)$

Theorem 1.1: for every $b\ge6$ and every finite-dimensional complex $V$ there is a $\mathrm{GL}(V)$-equivariant injection $\mathrm{Sym}^6(\mathrm{Sym}^bV)\hookrightarrow\mathrm{Sym}^b(\mathrm{Sym}^6V)$, i.e. $h_b[h_6]-h_6[h_b]$ is Schur-positive. The range $b\ge30$ comes from the companion's quadratic stabilization theorem; $6\le b\le29$ is settled by certified comparisons of upper bounds for source multiplicities with lower bounds for target multiplicities. It proves the multiplicity comparison, not injectivity of the canonical map (which fails at $(6,6)$), and only the case $a=6$; Foulkes' conjecture for $a\ge7$ and $b<a(a-1)$ stays open.

Theorem 1.1: for every $b\ge6$ and every finite-dimensional complex $V$ there is a $\mathrm{GL}(V)$-equivariant injection $\mathrm{Sym}^6(\mathrm{Sym}^bV)\hookrightarrow\mathrm{Sym}^b(\mathrm{Sym}^6V)$, i.e. $h_b[h_6]-h_6[h_b]$ is Schur-positive. The range $b\ge30$ comes from the companion's quadratic stabilization theorem; $6\le b\le29$ is settled by certified comparisons of upper bounds for source multiplicities with lower bounds for target multiplicities. It proves the multiplicity comparison, not injectivity of the canonical map (which fails at $(6,6)$), and only the case $a=6$; Foulkes' conjecture for $a\ge7$ and $b<a(a-1)$ stays open.