Acta Mathematica Vietnamica

, Volume 44, Issue 3, pp 603–615 | Cite as

Jacobi-Trudi Type Formula for Character of Irreducible Representations of \(\frak {gl}(m|1)\)

  • Nguyên Luong Thái Bình
  • Nguyên Thi Phuong Dung
  • Phùng Hô HaiEmail author


We prove a determinantal type formula to compute the irreducible characters of the general Lie superalgebra \(\mathfrak {gl}(m|1)\) in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula was conjectured by J. van der Jeugt and E. Moens for the Lie superalgebra \(\frak {gl}(m|n)\) and generalizes the well-known Jacobi-Trudi formula.


Character formula Jacobi-trudi type formula Character formula of Lie superalgebra 

Mathematics Subject Classification (2010)

17B10 17B15 17B20 17B22 



The authors would like to thank VIASM for the financial support and the excellent working environment.

Funding Information

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED), grant number 101.04-2016.19. A part of this work was carried out when the first and the third named authors were visiting the Vietnam Institute for Advanced Study in Mathematics.


  1. 1.
    Berele, A., Regev, A.: Hook Young diagrams with applications to combinatorics and to representation of Lie superalgebras. Adv. Math. 64(2), 118–175 (1987)CrossRefzbMATHGoogle Scholar
  2. 2.
    Brundan, J.: Kazhdan-lusztig polynomials and character formulae for the Lie superalgebra \(\mathfrak {gl}(m|n)\). J. Am. Math. Soc. 16(1), 185–231 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Balantekin, A.B., Bars, I.: Dimension and character formulas for Lie supergrous. J. Math. Phys. 22(6), 1149–1162 (1981)CrossRefzbMATHGoogle Scholar
  4. 4.
    Balantekin, A.B., Bars, I.: Representation of supergrous. J. Math. Phys. 22 (8), 1810–1818 (1981)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cummins, C.J., King, R.C.: Composite Young diagrams, supercharacters of U(M/N) and modification rules. J. Phys. A 20(11), 3121–3133 (1987)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dondi, P.H., Jarvis, P.D.: Diagram and superfield techniques in the classical superalgebras. J. Phys. A 14(3), 547–563 (1981)CrossRefzbMATHGoogle Scholar
  7. 7.
    Dung, N.T.P., Hai, P.H., Hung, N.H.: Construction of irreducible representations of the quantum super group G L q(3|1). Acta. Math. Vietnam. 36(2), 215–229 (2011)zbMATHGoogle Scholar
  8. 8.
    Dung, N.T.P.: Double Koszul complex and construction of irreducible representations of \(\mathfrak {gl}(3|1)\). Proc. Am. Math. Soc. 138(11), 3783–3796 (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    Dung, N.T.P., Hai, P.H.: Irreducible representations of quantum linear groups of type a 1|0. J. Algebra 282(2), 809–830 (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hughes, J.W.B., King, R.C., Van der Jeugt, J.: On the composition factors of Kac modules for the Lie superalgebras \(\frak {sl(m/n)}\). J. Math. Phys. 33(2), 470–491 (1992)CrossRefGoogle Scholar
  11. 11.
    Kac, V.G.: Classification of simple Lie superalgebras. Funct. Anal. Appl. 9(3), 91–92 (1975)zbMATHGoogle Scholar
  12. 12.
    Kac, V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kac, V.G.: Character of typical representations of classical Lie superalgebras. Comm. Algebra 5(8), 889–897 (1977)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kac, V.G.: Representations of Classical Lie Superalgebras. In: Lecture Notes in Math, vol. 676, pp. 597–626. Springer, Berlin (1978)Google Scholar
  15. 15.
    Macdonald, I.G.: Symmetric Function and Hall Polynomials. Oxford University Press, New York (1979)zbMATHGoogle Scholar
  16. 16.
    Moens, E.M., Van der Jeugt, J.: A determinantal fomula for supersymmetric Schur polynomials. J. Algebraic Combin. 17(3), 283–307 (2003)CrossRefzbMATHGoogle Scholar
  17. 17.
    Moens, E.M., Van der Jeugt, J.: On dimension formulas for \(\mathfrak {gl}(m|n)\) representations. J. Lie Theory 14(2), 523–535 (2004)zbMATHGoogle Scholar
  18. 18.
    Moens, E.M., Van der Jeugt, J.: On characters and dimension fomulas for representations of the Lie superalgebra \(\mathfrak {gl}(m|n)\). In: Doebner, H.-D., Dobrev, V.K. (eds.) Lie Theory and its Applications in Physics V, pp. 64–73. World Sci. Publ., River Edge (2004)Google Scholar
  19. 19.
    Moens, E.M., Van der Jeugt, J.: A character formula for atypical critical \(\mathfrak {gl}(m|n)\) representations labelled by composite partitions. J. Phys. A 37(50), 12019–12039 (2004)CrossRefzbMATHGoogle Scholar
  20. 20.
    Moens, E.M., Van der Jeugt, J.: Composite super-symmetric S-functions and character of \(\mathfrak {gl}(m|n)\) representations. In: Doebner, H.-D., Dobrev, V.K. (eds.) Proceedings of the VI International Worshop on Lie Theory and its Applications in Physics, pp. 251–268. Heron Press Ltd, Sofia (2006)Google Scholar
  21. 21.
    Moens, E.M.: Supersymmetric Schur Functions and Lie Superalgebra Representations. University of Gent, Ph.D. thesis (2006)Google Scholar
  22. 22.
    Su, Y., Zhang, R.B.: Character and dimension formulae for general linear superalgebra. Adv. Math. 211(1), 1–33 (2007)CrossRefzbMATHGoogle Scholar
  23. 23.
    Van der Jeugt, J., Hughes, J.W.B., King, R.C., Thierry-Mieg, J.: Character formulas for irreducible modules of the Lie superalgebras \(\mathfrak {sl}(m/n)\). J. Math. Phys. 31(9), 2278–2304 (1990)CrossRefzbMATHGoogle Scholar
  24. 24.
    Van der Jeugt, J., Hughes, J.W.B., King, R.C., Thierry-Mieg, J.: A character fomula for singly atypical modules of the Lie superalgebra \(\mathfrak {sl}(m/n)\). Comm. Algebra 18(10), 3453–3480 (1990)CrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Nguyên Luong Thái Bình
    • 1
  • Nguyên Thi Phuong Dung
    • 2
  • Phùng Hô Hai
    • 3
    Email author
  1. 1.Sai Gon UniversityHo Chi Minh CityVietnam
  2. 2.Banking AcademyHanoiVietnam
  3. 3.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

Personalised recommendations