GAUSS MAPS AND PLETHYSM

The irreducible representations of Gl(n, C) can be described by Schur functors, the composition of which defines plethysm. Its understanding is an important problem of invariant theory, as well as in relation w ith the representations of symmetric groups. In this paper, we address the problem geometrically. Through a generalization of the classical Veronese or Segre embeddings, we construct embeddings of flag manifold s into other flag manifolds, on which plethysm can be interpreted in t erms of sections of suitable line bundles. We infer the existence of n atural filtrations of plethysm, which readily implies different proper ties of its multiplicities: Vanishing conditions, growth, asymptotic b ehavior. In particular, we discuss the possibility to describe, thanks to our construction, the moment-polytopes attached to the asymptotics of plethysm.