Character and dimension formulae for general linear superalgebra

2004/03/19 by Yucai Su, Su, Yucai, R. B. Zhang +1 · 2 citations
Mathematics · #17B10 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT #msc:17B10

paper · pdf · doi:10.48550/arxiv.math/0403315

28 pages, re-writing results in terms of height vectors and adding one more result

openalex publication_date 2004/03/19 · arxiv created 2004/05/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The generalized Kazhdan-Lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly. Using the result we establish a one to one correspondence between the set of composition factors of an arbitrary r-fold atypical gl_m|n-Kac-module and the set of composition factors of some r-fold atypical gl_r|r-Kac-module. The result of Kazhdan-Lusztig polynomials is also applied to prove a conjectural character formula put forward by van der Jeugt et al in the late 80s. We simplify this character formula to cast it into the Kac-Weyl form, and derive from it a closed formula for the dimension of any finite dimensional irreducible representation of the general linear superalgebra.

Cited by

• Lie Superalgebras and Generalized Kazhdan-Lusztig Polynomials
• \mathfrak b₁-Verma \mathfrak b₂-dual Verma supermodules

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