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Irreducible Characters of General Linear Superalgebra and Super Duality

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Abstract

We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finite-dimensional modules, by directly relating the problem to the classical Kazhdan-Lusztig theory. Furthermore, we prove that certain parabolic BGG categories over the general linear algebra and over the general linear superalgebra are equivalent. We also verify a parabolic version of a conjecture of Brundan on the irreducible characters in the BGG category of the general linear superalgebra.

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References

  1. Brundan J.: Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra \({\mathfrak{gl}(m|n)}\). J. Amer. Math. Soc. 16, 185–231 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Beilinson A., Bernstein J.: Localisation de \({\mathfrak g}\)-modules. C.R. Acad. Sci. Paris Ser. I Math. 292, 15–18 (1981)

    MATH  MathSciNet  Google Scholar 

  3. Brylinski J.L., Kashiwara M.: Kazhdan-Lusztig conjecture and holonomic systerms. Invent. Math. 64, 387–410 (1981)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Berele A., Regev A.: Hook Young diagrams with applications to combinatorics and representations of Lie superalgebras. Adv. Math. 64, 118–175 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebras I: cellularity. http://arxiv.org/abs/0806.1532v1[math.RT], 2008

  6. Cheng, S.-J., Kwon, J.-H.: Howe duality and Kostant’s homology formula for infinite-dimensional Lie superalgebras. Int. Math. Res. Not. 2008, Art. ID rnn 085 (2008), 52 pp

  7. Cheng S.-J., Kwon J.-H., Lam N.: A BGG-type resolution for tensor modules over general linear superalgebra. Lett. Math. Phys. 84, 75–87 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Cheng, S.-J., Lam, N., Wang, W.: Super duality and irreducible characters of ortho-symplectic Lie superalgebras. http://arxiv.org/abs/0911.0129v1[math.RT], 2009

  9. Cheng S.-J., Lam N., Zhang R.B.: Character Formula for Infinite Dimensional Unitarizable Modules of the General Linear Superalgebra. J. Algebra 273, 780–805 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cheng S.-J., Wang W.: Brundan-Kazhdan-Lusztig and Super Duality Conjectures. Publ. Res. Inst. Math. Sci. 44, 1219–1272 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cheng S.-J., Wang W., Zhang R.B.: Super duality and Kazhdan-Lusztig polynomials. Trans. Amer. Math. Soc. 360, 5883–5924 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Deodhar V.: On some geometric aspects of Bruhat orderings II: the parabolic analogue of Kazhdan-Lusztig polynomials. J. Algebra 111, 483–506 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. Enright T., Willenbring J.: Hilbert series, Howe duality and branching for classical groups. Ann. of Math. (2) 159, 337–375 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Garland H., Lepowsky J.: Lie Algebra Homology and the Macdonald-Kac Formulas. Invent. Math. 34, 37–76 (1976)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Humphreys, J.: Representations of Semisimple Lie Algebras in the BGG Category \({\mathcal {O}}\). Graduate Studies in Mathematics 94, Providence, RI: Amer. Math. Soc. 2008

  16. Hochschild G.: Relative homological algebra. Trans. Amer. Math. Soc. 82, 246–269 (1956)

    MATH  MathSciNet  Google Scholar 

  17. Kac V.: Lie superalgebras. Adv. Math. 16, 8–96 (1977)

    Article  Google Scholar 

  18. Kac, V.: Representations of Classical Lie Superalgebras. Lect. Notes in Math. 676, Berlin Heidelberg-New york: Springer Verlag, 1978, pp. 597–626

  19. Kumar, S.: Kac-Moody Groups, their Flag Varieties and Representation Theory. Progress in Mathematics, 204. Boston, MA: Birkhauser Boston, Inc., 2002

  20. Kac V., Wakimoto M.: Integrable highest weight modules over affine superalgebras and Appell’s function. Commun. Math. Phys. 215, 631–682 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Kazhdan D., Lusztig G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1979)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Liu L.: Kostant’s Formula for Kac-Moody Lie algebras. J. Algebra 149, 155–178 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lepowsky J.: A generalization of the Bernstein-Gelfand-Gelfand resolution. J. Algebra 49, 496–511 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  24. Leites, D., Saveliev, M., Serganova, V.: Embedding of osp(N/2) and the associated non-linear supersymmetric equations. In: Group Theoretical Methods in Physics, Vol. I (Yurmala, 1985), Utrecht: VNU Sci. Press, 1986, pp. 255–297

  25. Lascoux A., Leclerc B., Thibon J.-Y.: Hecke algebras at roots of unity and crystal bases of quantum affine algebras. Commun. Math. Phys. 181, 205–263 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Mitchell, B.: Theory of Categories. Pure and Applied Mathematics XVII, New York-San Francisco-London: Academic Press, 1965

  27. Popescu, N.: Abelian Categories with Applications to Rings and Modules. London Mathematical Society Monographs 3, London-New York: Academic Press, 1973

  28. Penkov I., Serganova V.: Cohomology of G/P for classical complex Lie supergroups G and characters of some atypical G-modules. Ann. Inst. Fourier 39, 845–873 (1989)

    MATH  MathSciNet  Google Scholar 

  29. Rocha-Caridi A., Wallach N.: Projective modules over graded Lie algebras I. Math. Z. 180, 151–177 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  30. Sergeev A.: The Tensor algebra of the identity representation as a module over the Lie superalgebras gl(n,m) and Q(n). Math. USSR Sbornik 51, 419–427 (1985)

    Article  MATH  Google Scholar 

  31. Serganova V.: Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra \({\mathfrak{gl}(m\vert n)}\). Selecta Math. (N.S.) 2, 607–651 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  32. Tanaka J.: On homology and cohomology of Lie superalgebras with coefficients in their finite-dimensional representations. Proc. Japan Acad. Ser. A Math. Sci. 71(3), 51–53 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  33. Vogan D.: Irreducible characters of semisimple Lie Groups II: The Kazhdan-Lusztig Conjectures. Duke Math. J. 46, 805–859 (1979)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Academia Sinica, Taipei, 10617, Taiwan

    Shun-Jen Cheng

  2. Department of Mathematics, National Cheng-Kung University, Tainan, 70101, Taiwan

    Ngau Lam

Authors
  1. Shun-Jen Cheng
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  2. Ngau Lam
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Correspondence to Shun-Jen Cheng.

Additional information

Communicated by Y. Kawahigashi

Partially supported by an NSC-grant and an Academia Sinica Investigator grant.

Partially supported by an NSC-grant.

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Cheng, SJ., Lam, N. Irreducible Characters of General Linear Superalgebra and Super Duality. Commun. Math. Phys. 298, 645–672 (2010). https://doi.org/10.1007/s00220-010-1087-7

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