Landsberg's Problem 7.19: a polynomial stabilization bound for the canonical Foulkes-Howe map

Theorem 1: for every $a\ge2$, $b\ge a(a-1)$ and finite-dimensional complex $V$, $\mu_{a,b,V}$ is surjective; equivalently the Chow-variety coordinate ring agrees with its normalization in degree $b$. In Landsberg's notation the bound is $d\ge n(n-1)$, in every dimension. By complete reducibility this yields Foulkes' embedding $\mathrm{Sym}^a(\mathrm{Sym}^bV)\hookrightarrow\mathrm{Sym}^b(\mathrm{Sym}^aV)$ for $b\ge a(a-1)$. The least stabilization degree is not determined, the bound is not claimed sharp, and nothing is said about injectivity of the map in the other direction (which fails at $(5,5)$ and $(6,6)$).

Theorem 1: for every $a\ge2$, $b\ge a(a-1)$ and finite-dimensional complex $V$, $\mu_{a,b,V}$ is surjective; equivalently the Chow-variety coordinate ring agrees with its normalization in degree $b$. In Landsberg's notation the bound is $d\ge n(n-1)$, in every dimension. By complete reducibility this yields Foulkes' embedding $\mathrm{Sym}^a(\mathrm{Sym}^bV)\hookrightarrow\mathrm{Sym}^b(\mathrm{Sym}^aV)$ for $b\ge a(a-1)$. The least stabilization degree is not determined, the bound is not claimed sharp, and nothing is said about injectivity of the map in the other direction (which fails at $(5,5)$ and $(6,6)$).