Geometric Invariant Theory and Generalized Eigenvalue Problem II

Ressayre, Nicolas

Annales de l'Institut Fourier, Tome 61 (2011), p. 1467-1491 / Harvested from Numdam

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Résumé
Abstract

Soit $G$ un sous-groupe fermé réductif et connexe d’un groupe réductif complexe et connexe $\hat{G}$ . On fixe des tores maximaux et des sous-groupes de Borel de $G$ et $\hat{G}$ . De cette manière les représentations irréductibles de $G$ et $\hat{G}$ sont paramétrées par des poids dominants. On s’intéresse au cône ${ℒ R}^{\circ} (G, \hat{G})$ engendré par les paires $(ν, \hat{ν})$ de poids dominants réguliers tels que $V_{ν}^{*}$ est un sous- $G$ -module de $V_{\hat{ν}}$ . Nous obtenons ici une paramétrisation bijective des faces de ${ℒ R}^{\circ} (G, \hat{G})$ , en étudiant plus généralement les GIT-cônes des $G$ -variétés projectives. Nous montrons aussi comment les relations d’inclusions entre les faces de ${ℒ R}^{\circ} (G, \hat{G})$ se lisent sur notre paramétrisation.

Let $G$ be a connected reductive subgroup of a complex connected reductive group $\hat{G}$ . Fix maximal tori and Borel subgroups of $G$ and $\hat{G}$ . Consider the cone $ℒ ℛ^{\circ} (G, \hat{G})$ generated by the pairs $(ν, \hat{ν})$ of strictly dominant characters such that $V_{ν}^{*}$ is a submodule of $V_{\hat{ν}}$ . We obtain a bijective parametrization of the faces of $ℒ ℛ^{\circ} (G, \hat{G})$ as a consequence of general results on GIT-cones. We show how to read the inclusion of faces off this parametrization.

Publié le : 2011-01-01

MR 2951500 | Zbl 1245.14045

DOI : https://doi.org/10.5802/aif.2647
Classification: 20G05, 14L24
Mots clés: problème de restriction, problème de Horn et ses généralisations, cône de Littlewood-Richardson, GIT- cône

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     author = {Ressayre, Nicolas},
     title = {Geometric Invariant Theory and Generalized Eigenvalue Problem II},
     journal = {Annales de l'Institut Fourier},
     volume = {61},
     year = {2011},
     pages = {1467-1491},
     doi = {10.5802/aif.2647},
     zbl = {1245.14045},
     mrnumber = {2951500},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2011__61_4_1467_0}
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Ressayre, Nicolas. Geometric Invariant Theory and Generalized Eigenvalue Problem II. Annales de l'Institut Fourier, Tome 61 (2011) pp. 1467-1491. doi : 10.5802/aif.2647. http://gdmltest.u-ga.fr/item/AIF_2011__61_4_1467_0/

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