Applications de Gauss et pléthysme
Manivel, Laurent
Annales de l'Institut Fourier, Tome 47 (1997), p. 715-773 / Harvested from Numdam
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Les représentations irréductibles de Gl (n,ℂ) sont décrites par les foncteurs de Schur, dont la composition définit le pléthysme. Sa compréhension est un problème important en théorie des invariants, ou bien en relation avec les représentations des groupes symétriques.

Nous proposons dans cet article une approche géométrique du problème. Généralisant les plongements classiques de Veronese et de Segre, nous construisons des plongements de variétés de drapeaux dans d’autres variétés de drapeaux, sur lesquels le pléthysme s’interprète en termes de sections de certains fibrés en droites. Nous en déduisons des filtrations naturelles, dont découlent différentes propriétés des multiplicités : conditions d’annulation, croissance, comportement asymptotique.

En particulier, nous discutons en détail la possibilité de décrire, grâce à nos plongements, les polytopes-moment associés à l’asymptotique du pléthysme.

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 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.

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     author = {Manivel, Laurent},
     title = {Applications de Gauss et pl\'ethysme},
     journal = {Annales de l'Institut Fourier},
     volume = {47},
     year = {1997},
     pages = {715-773},
     doi = {10.5802/aif.1579},
     mrnumber = {98h:20078},
     zbl = {0868.05054},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1997__47_3_715_0}
}

Manivel, Laurent. Applications de Gauss et pléthysme. Annales de l'Institut Fourier, Tome 47 (1997) pp. 715-773. doi : 10.5802/aif.1579. http://gdmltest.u-ga.fr/item/AIF_1997__47_3_715_0/

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