  Let $s_\nu \circ s_\mu$ denote the plethystic product of the Schur functions
$s_\nu$ and $s_\mu$. In this article we define an explicit polynomial
representation corresponding to $s_\nu \circ s_\mu$ with basis indexed by
certain `plethystic' semistandard tableaux. Using these representations we
prove generalizations of four results on plethysms due to
Bruns--Conca--Varbaro, Brion, Ikenmeyer and the authors. In particular, we give
a sufficient condition for the multiplicity $\langle s_\nu \circ s_\mu,
s_\lambda\rangle$ to be stable under insertion of new parts into $\mu$ and
$\lambda$. We also characterize all maximal and minimal partitions $\lambda$ in
the dominance order such that $s_\lambda$ appears in $s_\nu \circ s_\mu$ and
determine the corresponding multiplicities using plethystic semistandard
tableaux.
