Let be a connected reductive subgroup of a complex connected reductive group . Fix maximal tori and Borel subgroups of and . Consider the cone generated by the pairs of strictly dominant characters such that is a submodule of . We obtain a bijective parametrization of the faces of as a consequence of general results on GIT-cones. We show how to read the inclusion of faces off this parametrization.
Soit un sous-groupe fermé réductif et connexe d’un groupe réductif complexe et connexe . On fixe des tores maximaux et des sous-groupes de Borel de et . De cette manière les représentations irréductibles de et sont paramétrées par des poids dominants. On s’intéresse au cône engendré par les paires de poids dominants réguliers tels que est un sous--module de . Nous obtenons ici une paramétrisation bijective des faces de , en étudiant plus généralement les GIT-cônes des -variétés projectives. Nous montrons aussi comment les relations d’inclusions entre les faces de se lisent sur notre paramétrisation.
Keywords: Branching rule, generalized Horn problem, Littlewood-Richardson cone, GIT-cone
Keywords: problème de restriction, problème de Horn et ses généralisations, cône de Littlewood-Richardson, GIT- cône
@article{AIF_2011__61_4_1467_0, author = {Ressayre, Nicolas}, title = {Geometric {Invariant} {Theory} and {Generalized} {Eigenvalue} {Problem} {II}}, journal = {Annales de l'Institut Fourier}, pages = {1467--1491}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {4}, year = {2011}, doi = {10.5802/aif.2647}, zbl = {1245.14045}, mrnumber = {2951500}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2647/} }
TY - JOUR AU - Ressayre, Nicolas TI - Geometric Invariant Theory and Generalized Eigenvalue Problem II JO - Annales de l'Institut Fourier PY - 2011 SP - 1467 EP - 1491 VL - 61 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2647/ DO - 10.5802/aif.2647 LA - en ID - AIF_2011__61_4_1467_0 ER -
%0 Journal Article %A Ressayre, Nicolas %T Geometric Invariant Theory and Generalized Eigenvalue Problem II %J Annales de l'Institut Fourier %D 2011 %P 1467-1491 %V 61 %N 4 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2647/ %R 10.5802/aif.2647 %G en %F AIF_2011__61_4_1467_0
Ressayre, Nicolas. Geometric Invariant Theory and Generalized Eigenvalue Problem II. Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1467-1491. doi : 10.5802/aif.2647. http://www.numdam.org/articles/10.5802/aif.2647/
[1] Geometric proof of a conjecture of Fulton, Adv. Math., Volume 216 (2007) no. 1, pp. 346-357 | DOI | MR | Zbl
[2] Eigenvalue problem and a new product in cohomology of flag varieties, Invent. Math., Volume 166 (2006) no. 1, pp. 185-228 | DOI | MR | Zbl
[3] Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion, J. Amer. Math. Soc., Volume 13 (2000) no. 2, pp. 433-466 | DOI | MR | Zbl
[4] On the general faces of the moment polytope, Internat. Math. Res. Notices (1999) no. 4, pp. 185-201 | DOI | MR | Zbl
[5] Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math., Volume 87 (1998), pp. 5-56 (With an appendix by Nicolas Ressayre) | DOI | MR | Zbl
[6] Eigenvalues of sums of Hermitian matrices, Pacific J. Math., Volume 12 (1962), pp. 225-241 | MR | Zbl
[7] The honeycomb model of tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc., Volume 17 (2004) no. 1, pp. 19-48 | DOI | MR | Zbl
[8] Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, p. 81-105. Bull. Soc. Math. France, Paris, Mémoire 33 | Numdam | MR | Zbl
[9] Adhérences d’orbite et invariants, Invent. Math., Volume 29 (1975) no. 3, pp. 231-238 | DOI | MR | Zbl
[10] A generalization of the Chevalley restriction theorem, Duke Math. J., Volume 46 (1979) no. 3, pp. 487-496 | DOI | MR | Zbl
[11] Geometric Invariant Theory, Springer Verlag, New York, 1994 | MR | Zbl
[12] Algebraic Geometry IV, Invariant Theory (Encyclopedia of Mathematical Sciences), Volume 55, Springer-Verlag, 1991, pp. 123-284
[13] A short geometric proof of a conjecture of Fulton à paraître dans Enseign. Math. (2)
[14] The GIT-equivalence for -line bundles, Geom. Dedicata, Volume 81 (2000) no. 1-3, pp. 295-324 | DOI | MR | Zbl
[15] Geometric Invariant Theory and Generalized Eigenvalue Problem, Invent. Math., Volume 180 (2010), pp. 389-441 | DOI | MR | Zbl
[16] Convexity properties of the moment mapping re-examined, Adv. Math., Volume 138 (1998) no. 1, pp. 46-91 | DOI | MR | Zbl
Cited by Sources: