PLETHYSMS OF SYMMETRIC FUNCTIONS AND HIGHEST WEIGHT REPRESENTATIONS

Let s(nu) circle 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) circle 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 < s(nu) circle s(mu) ,s(lambda)> 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) o s(mu) and determine the corresponding multiplicities using plethystic semistandard tableaux.