Letters in Mathematical Physics

, Volume 47, Issue 1, pp 49–61 | Cite as

En">Characters and Composition Factor Multiplicities for the Lie Superalgebra \({\mathfrak{g}}{\mathfrak{l}}\) ( m / n )

  • J. Van der Jeugt
  • R. B. Zhang


The multiplicities aλ μ of simple modules Lμ in the composition series of Kac modules V lambda for the Lie superalgebra \({\mathfrak{g}}{\mathfrak{l}}\) (m/n ) were described by Serganova, leading to her solution of the character problem for \({\mathfrak{g}}{\mathfrak{l}}\) (m/n ). In Serganova's algorithm all μ with nonzero aλ μ are determined for a given λ this algorithm, turns out to be rather complicated. In this Letter, a simple rule is conjectured to find all nonzero aλ μ for any given weight μ. In particular, we claim that for an r-fold atypical weight μ there are 2r distinct weights λ such that aλ μ = 1, and aλ μ = 0 for all other weights λ. Some related properties on the multiplicities aλ μ are proved, and arguments in favour of our main conjecture are given. Finally, an extension of the conjecture describing the inverse of the matrix of Kazhdan–Lusztig polynomials is discussed.

character formula Lie superalgebra \({\mathfrak{g}}{\mathfrak{l}}\) (m/n ) Kac module composition factors Kazhdan–Lusztig polynomial. 


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  1. 1.
    Berele, A. and Regev, A.: Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math. 64(2) (1987), 118–175.Google Scholar
  2. 2.
    Bernstein, I. N. and Leites, D. A.: Character formulae for irreducible representations of Lie superalgebras of series gl and sl, C.R. Acad. Bulg. Sci. 33 (1980), 1049–51.Google Scholar
  3. 3.
    Dondi, P. H. and Jarvis, P. D.: Diagram and superfield techniques in the classical superalgebras, J. Phys. A 14 (1981), 547–563.Google Scholar
  4. 4.
    Hughes, J.W. B., King, R. C. and Van der Jeugt, J.: On the composition factors of Kac modules for the Lie superalgebra sl(m/n), J. Math. Phys. 33 (1992), 470–491.Google Scholar
  5. 5.
    Kac, V. G.: Lie superalgebras, Adv. in Math. 26 (1977), 8–96.Google Scholar
  6. 6.
    Kac, V. G.: Representations of classical Lie superalgebras, in Lecture Notes in Math. 676, Springer, New York, 1978, pp. 597–626.Google Scholar
  7. 7.
    Kac, V. G. and Wakimoto, M.: Integrable highest weight modules over affine superalgebras and number theory, Progr. Math. 123 (1994), 415–456.Google Scholar
  8. 8.
    Penkov, I. and Serganova, V.: On irreducible representations of classical Lie superalgebras, Indag. Math. 3 (1992), 419–466.Google Scholar
  9. 9.
    Penkov, I. and Serganova, V.: Generic irreducible representations of finite-dimensional Lie superalgebras, Internat. J. Math. 5 (1994), 389–419.Google Scholar
  10. 10.
    Penkov, I. and Serganova, V.: Characters of finite-dimensional irreducible q(n)-modules, Lett. Math. Phys. 40 (1997), 147–158.Google Scholar
  11. 11.
    Scheunert, M.: The Theory of Lie Superalgebras, Springer, Berlin, 1979.Google Scholar
  12. 12.
    Serganova, V.: Kazhdan-Lusztig polynomials for Lie superalgebra GL(m, n), Adv. Soviet Math. 16 (1993), 151–165.Google Scholar
  13. 13.
    Serganova, V.: Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra gl, Selecta Math. 2(4) (1996), 607–651.Google Scholar
  14. 14.
    Van der Jeugt, J., Hughes, J. W. B., King, R. C. and Thierry-Mieg, J.: Character formulas for irreducible modules of the Lie superalgebra sl(m/n), J. Math. Phys. 31 (1990), 2278–2304.Google Scholar
  15. 15.
    Van der Jeugt, J., Hughes, J. W. B., King, R. C. and Thierry-Mieg, J.: A character formula for singly atypical modules of the Lie superalgebra sl(m/n), Comm. Algebra 18 (1990), 3453–3480.Google Scholar
  16. 16.
    Van der Jeugt, J.: Character formulae for the Lie superalgebra C(n), Comm. Algebra 19 (1991), 199–222.Google Scholar
  17. 17.
    Zou, Y.M.: Categories of finite dimensional weight modules over type I classical Lie superalgebras, J. Algebra 180 (1996), 459–482.Google Scholar

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© Kluwer Academic Publishers 1999

Authors and Affiliations

  • J. Van der Jeugt
  • R. B. Zhang

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