2013/12/02 by Jérémie Guilhot, Guilhot, Jérémie, Cédric Lecouvey +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1312.0470
15 pages
arxiv created 2013/12/02 · openalex publication_date 2013/12/02 · arxiv updated 2013/12/03 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Consider a simple Lie algebra \mathfrakg and \mathfrakg% ⊂ \mathfrakg a Levi subalgebra. Two irreducible % \mathfrakg-modules yield isomorphic inductions to \mathfrakg when their highest weights coincide up to conjugation by an element of the Weyl group W of \mathfrakg which is also a Dynkin diagram automorphism of % \mathfrakg. In this paper we study the converse problem: given two irreducible \mathfrakg-modules of highest weight μ and ν whose inductions to \mathfrakg 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 \mathfrakg% ? 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 coefficient of the main result of \citeRaj on tensor product multiplicities.