Positivity of vector bundles and Dominance

Let $E$ be a vector bundle and $S_a$, $S_b$ the Schur functors associated to partitions $a$ and $b$. Previously we have shown that ampleness of $S_aE$ implies ampleness of $S_bE$ when $a$ is greater than $b$ in the dominance partial order. Here we prove that this result generalizes to $k$-ample, semiample and nef vector bundles. Our proof uses the common algebraic nature of these three properties and an investigation of the Littlewood-Richardson rules.