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Gauss maps and plethysm. (Applications de Gauss et pléthysme.) (French) Zbl 0868.05054
Summary: The irreducible representations of \(\text{Gl}(n,\mathbb{C})\) 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 with 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 manifolds into other flag manifolds, on which plethysm can be interpreted in terms of sections of suitable line bundles. We infer the existence of natural filtrations of plethysm, which readily implies different properties of its multiplicities: vanishing conditions, growth, asymptotic behavior. In particular, we discuss the possibility to describe, thanks to our construction, the moment-polytopes attached to the asymptotics of plethysm.

MSC:
05E10 Combinatorial aspects of representation theory
14M15 Grassmannians, Schubert varieties, flag manifolds
20G05 Representation theory for linear algebraic groups
20C30 Representations of finite symmetric groups
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