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Two generalizations of the PRV conjecture

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

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: pierre-louis.montagard@math.univ-montp2.fr)
B. Pasquier
Affiliation:
Université Montpellier II, CC 51-Place Eugène Bataillon, 34095 Montpellier Cedex 5, France (email: boris.pasquier@math.univ-montp2.fr)
N. Ressayre
Affiliation:
Université Montpellier II, CC 51-Place Eugène Bataillon, 34095 Montpellier Cedex 5, France (email: nicolas.ressayre@math.univ-montp2.fr)
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Abstract

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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

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