On the irreducibility of irreducible characters of simple Lie algebras

2011/10/24 by C. S. Rajan, Rajan, C. S.
Mathematics · #17B10 #20G05 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:17B10 #msc:20G05

paper · pdf · doi:10.48550/arxiv.1110.5300

48 pages

arxiv created 2011/10/24 · openalex publication_date 2011/10/24 · arxiv updated 2011/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors. For SL(r) the irreducibility result is the following: let λ=(a₁≥ a₂≥ ... a_r-1≥ 0) be the highest weight of an irreducible rational representation V_λ of SL(r). Assume that the integers a₁+r-1, ~a₂+r-2,..., a_r-1+1 are relatively prime. Then the character χ_λ of V_λ is strongly irreducible in the following sense: for any natural number d, the function χ_λ(g^d), ~g∈ SL(r,\C) is irreducible in the ring of regular functions of SL(r,\C).

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