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ISOMORPHIC INDUCED MODULES AND DYNKIN DIAGRAM AUTOMORPHISMS OF SEMISIMPLE LIE ALGEBRAS

Part of: Lie algebras and Lie superalgebras Algebraic combinatorics

Published online by Cambridge University Press:  21 July 2015

JÉRÉMIE GUILHOT  and
CÉDRIC LECOUVEY
JÉRÉMIE GUILHOT
Affiliation:
Laboratoire de Mathématiques et Physique Théorique, (UMR CNRS 7350) Université François-Rabelais, Tours Fédération de Recherche Denis Poisson. e-mail: jeremie.guilhot@lmpt.univ-tours.fr; cedric.lecouvey@lmpt.univ-tours.fr
CÉDRIC LECOUVEY
Affiliation:
Laboratoire de Mathématiques et Physique Théorique, (UMR CNRS 7350) Université François-Rabelais, Tours Fédération de Recherche Denis Poisson. e-mail: jeremie.guilhot@lmpt.univ-tours.fr; cedric.lecouvey@lmpt.univ-tours.fr
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Abstract

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Consider a simple Lie algebra $\mathfrak{g}$ and $\overline{\mathfrak{g}}$ ⊂ $\mathfrak{g}$ a Levi subalgebra. Two irreducible $\overline{\mathfrak{g}}$-modules yield isomorphic inductions to $\mathfrak{g}$ when their highest weights coincide up to conjugation by an element of the Weyl group W of $\mathfrak{g}$ which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$. In this paper, we study the converse problem: given two irreducible $\overline{\mathfrak{g}}$-modules of highest weight μ and ν whose inductions to $\mathfrak{g}$ are isomorphic, can we conclude that μ and ν are conjugate under the action of an element of W which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$? We conjecture this is true in general. We prove this conjecture in type A and, for the other root systems, in various situations providing μ and ν satisfy additional hypotheses. Our result can be interpreted as an analogue for branching coefficients of the main result of Rajan [6] on tensor product multiplicities.

MSC classification

Primary: 05E05: Symmetric functions and generalizations
Secondary: 05E10: Combinatorial aspects of representation theory 17B20: Simple, semisimple, reductive (super)algebras 17B22: Root systems
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 1 , January 2016 , pp. 187 - 203
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

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