Descriptions of strongly multiplicity free representations for simple Lie algebras

2023/04/23 by Binni Sun, Yufeng Zhao, Sun, Binni +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2304.11601

openalex publication_date 2023/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathfrakg be a complex simple Lie algebra and Z(\mathfrakg) be the center of the universal enveloping algebra U(\mathfrakg). Denote by V_λ the finite-dimensional irreducible \mathfrakg-module with highest weight λ. Lehrer and Zhang defined the notion of strongly multiplicity free representations for simple Lie algebras motivated by studying the structure of the endomorphism algebra End_U(\mathfrakg)(V_λ^{⊗ r}) in terms of the quotients of the Kohno's infinitesimal braid algebra. Kostant introduced the \mathfrakg-invariant endomorphism algebras R_λ(\mathfrakg)= (End V_λ⊗ U(\mathfrakg))^\mathfrakg and R_λ,π(\mathfrakg)=(End V_λ⊗ π(U(\mathfrakg)))^\mathfrakg. In this paper, we give some other criteria for a multiplicity free representation to be strongly multiplicity free by classifying the pairs (\mathfrakg, V_λ), which are multiplicity free and for such pairs, R_λ(\mathfrakg) and R_λ,π(\mathfrakg) are generated by generalizations of the quadratic Casimir elements of Z(\mathfrakg).

Cited by

• Center of Kostant algebra
• Irreducibility of polarized automorphic Galois representations in infinitely many dimensions

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