Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2025-01-03T18:37:39.045Z Has data issue: false hasContentIssue false
  • English
  • Français

Two generalizations of the PRV conjecture

Part of: Lie groups Algebraic groups Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  18 March 2011

P. L. Montagard ,
B. Pasquier  and
N. Ressayre
P. L. Montagard
Affiliation:
Université Montpellier II, CC 51-Place Eugène Bataillon, 34095 Montpellier Cedex 5, France (email: [email protected])
B. Pasquier
Affiliation:
Université Montpellier II, CC 51-Place Eugène Bataillon, 34095 Montpellier Cedex 5, France (email: [email protected])
N. Ressayre
Affiliation:
Université Montpellier II, CC 51-Place Eugène Bataillon, 34095 Montpellier Cedex 5, France (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a complex connected reductive group. The Parthasarathy–Ranga Rao–Varadarajan (PRV) conjecture, which was proved independently by S. Kumar and O. Mathieu in 1989, gives explicit irreducible submodules of the tensor product of two irreducible G-modules. This paper has three aims. First, we simplify the proof of the PRV conjecture, then we generalize it to other branching problems. Finally, we find other irreducible components of the tensor product of two irreducible G-modules that appear for ‘the same reason’ as the PRV ones.

Keywords

tensor product decompositionbranching rulesPRV conjecture

MSC classification

Secondary: 22E46: Semisimple Lie groups and their representations 17B10: Representations, algebraic theory (weights) 14L24: Geometric invariant theory
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 4 , July 2011 , pp. 1321 - 1336
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[And85]Andersen, H. H., Schubert varieties and Demazure’s character formula, Invent. Math. 79 (1985), 611–618.CrossRefGoogle Scholar
[BK10]Belkale, P. and Kumar, S., Eigencone, saturation and horn problems for symplectic and odd orthogonal groups, J. Algebraic Geom. 19 (2010), 199–242.CrossRefGoogle Scholar
[Bri01]Brion, M., On orbit closures of spherical subgroups in flag varieties, Comment. Math. Helv. 76 (2001), 263–299.CrossRefGoogle Scholar
[Bri03]Brion, M., Multiplicity-free subvarieties of flag varieties, in Commutative algebra (Grenoble/Lyon, 2001), Contemporary Mathematics, vol. 331 (American Mathematical Society, Providence, RI, 2003), 13–23.CrossRefGoogle Scholar
[Dem74]Demazure, M., Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88.CrossRefGoogle Scholar
[FH91]Fulton, W. and Harris, J., Representation theory, Graduate Texts in Mathematics, vol. 129 (Springer, New York, 1991).Google Scholar
[Jos85]Joseph, A., On the Demazure character formula, Ann. Sci. École Norm. Sup. (4) 18 (1985), 389–419.CrossRefGoogle Scholar
[KLM08]Kapovich, M., Leeb, B. and Millson, J. J., The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra, Mem. Amer. Math. Soc. 192 (2008), viii and 83.Google Scholar
[KT99]Knutson, A. and Tao, T., The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), 1055–1090.CrossRefGoogle Scholar
[Kum88]Kumar, S., Proof of the Parthasarathy–Ranga Rao–Varadarajan conjecture, Invent. Math. 93 (1988), 117–130.CrossRefGoogle Scholar
[Kum92]Kumar, S., Proof of Wahl’s conjecture on surjectivity of the Gaussian map for flag varieties, Amer. J. Math. 114 (1992), 1201–1220.CrossRefGoogle Scholar
[Kum11]Kumar, S., Tensor product decomposition in Proceedings of the international congress of mathematicians 2010, Hyderabad, India (World Scientific, 2011), available at http://www.math.unc.edu/Faculty/kumar/papers/kumar60.pdf.CrossRefGoogle Scholar
[Lun75]Luna, D., Adhérences d’orbite et invariants, Invent. Math. 29 (1975), 231–238.CrossRefGoogle Scholar
[Mat89]Mathieu, O., Construction d’un groupe de Kac–Moody et applications, Compositio Math. 69 (1989), 37–60.Google Scholar
[MFK94]Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, third edition (Springer, New York, 1994).CrossRefGoogle Scholar
[RR85]Ramanan, S. and Ramanathan, A., Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), 217–224.CrossRefGoogle Scholar
[Res10a]Ressayre, N., Spherical homogeneous spaces of minimal rank, Adv. Math. 224 (2010), 1784–1800.CrossRefGoogle Scholar
[Res10b]Ressayre, N., Geometric invariant theory and generalized eigenvalue problem, Invent. Math. 180 (2010), 389–441.CrossRefGoogle Scholar
[Ros61]Rosenlicht, M., On quotient varieties and the affine embedding of certain homogeneous spaces, Trans. Amer. Math. Soc. 101 (1961), 211–223.CrossRefGoogle Scholar
[Ses87]Seshadri, C. S., Line bundles on Schubert varieties, in Vector bundles on algebraic varieties, Bombay, 1984, Tata Institute of Fundamental Research Studies in Mathematics, vol. 11 (Tata Institute of Fundamental Research Studies Bombay, Bombay, 1987), 499–528.Google Scholar
[Sun90]Sundaram, S., Tableaux in the representation theory of the classical Lie groups, in Invariant theory and tableaux, Minneapolis, MN, 1988, The IMA Volumes in Mathematics and its Applications, vol. 19 (Springer, New York, 1990), 191–225.Google Scholar