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An explicit formula for the characters of the symmetric group

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Abstract

We give an explicit expression of the normalized characters of the symmetric group in terms of the “contents” of the partition labelling the representation.

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References

  1. Biane P. (1998). Representations of symmetric groups and free probability. Adv. Math. 138: 126–181

    Article  MATH  MathSciNet  Google Scholar 

  2. Biane, P.: On the formula of Goulden and Rattan for Kerov polynomials. Sém. Lothar. Combin., 55, article B55d (2006)

  3. Corteel S., Goupil A. and Schaeffer G. (2004). Content evaluation and class symmetric functions. Adv. Math. 188: 315–336

    Article  MATH  MathSciNet  Google Scholar 

  4. Désarménien, J.: Une généralisation des caractères du groupe symétrique. Unpublished note (1996)

  5. Frobenius G.: Über die Charaktere der Symmetrischen Gruppe. Sützungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1900), pp. 516–534. Reprinted in Gessamelte Abhandlungen 3, 148–166

  6. Garsia, A.: Young seminormal representation, Murphy elements and content evaluations. Lecture notes (march 2003), available at http://www.math.ucsd.edu/~garsia/recentpapers/

  7. Goldschmidt, D.M.: Group characters, symmetric functions and the Hecke algebra. University Lecture Series 4. Am. Math. Soc., Providence (1991)

  8. Goulden, I.P., Rattan, A.: An explicit form for Kerov’s character polynomials. Trans. Am. Math. Soc. 359, 3669–3685 (2007)

    Google Scholar 

  9. Ingram R.E. (1950). Some characters of the symmetric group. Proc. Am. Math. Soc. 1: 358–369

    Article  MATH  MathSciNet  Google Scholar 

  10. Ivanov, V., Olshanski, G.I.: Kerov’s central limit theorem for the Plancherel measure on Young diagrams. In Symmetric functions 2001: Surveys of developments and perspectives, pp. 93–151. Kluwer, Dordrecht (2002)

  11. Jucys A.A. (1974). Symmetric polynomials and the center of the symmetric group ring. Rep. Math. Phys. 5: 107–112

    Article  MATH  MathSciNet  Google Scholar 

  12. Katriel J. (1996). Explicit expressions for the central characters of the symmetric group. Discrete Appl. Math. 67: 149–156

    Article  MATH  MathSciNet  Google Scholar 

  13. Kerov S.V. and Olshanski G.I. (1994). Polynomial functions on the set of Young diagrams. C. R. Acad. Sci. Paris Sér. I 319: 121–126

    MATH  MathSciNet  Google Scholar 

  14. Lascoux, A.: Notes on interpolation in one and several variables. Available at http://igm.univ-mlv.fr/~al/

  15. Lascoux A. and Thibon J.-Y. (2001). Vertex operators and the class algebras of symmetric groups. Zapiski Nauchnyh Seminarov POMI 283: 156–177

    Google Scholar 

  16. Lassalle M. (1998). Some combinatorial conjectures for Jack polynomials. Ann. Comb. 2: 61–83

    Article  MATH  MathSciNet  Google Scholar 

  17. Lassalle M. (2001). Une q-spécialisation pour les fonctions symétriques monomiales. Adv. Math. 162: 217–242

    Article  MATH  MathSciNet  Google Scholar 

  18. Lassalle M. (2002). A new family of positive integers. Ann. Comb. 6: 399–405

    MATH  MathSciNet  Google Scholar 

  19. Lassalle M. (2004). Jack polynomials and some identities for partitions. Trans. Am. Math. Soc. 356: 3455–3476

    Article  MATH  MathSciNet  Google Scholar 

  20. Lassalle M. (2005). Explicitation of characters of the symmetric group. C. R. Acad. Sci. Paris Sér. I 341: 529–534

    MATH  MathSciNet  Google Scholar 

  21. Lassalle, M.: Available at http://igm.univ-mlv.fr/~lassalle/char.html

  22. Macdonald I.G. (1995). Symmetric functions and Hall polynomials, 2nd edn. Clarendon Press, Oxford

    Google Scholar 

  23. Murnaghan F.D. (1937). On the representations of the symmetric group. Am. J. Math. 59: 739–753

    Article  MATH  MathSciNet  Google Scholar 

  24. Murphy G.E. (1981). A new construction of Young’s seminormal representation of the symmetric group. J. Algebra 69: 287–291

    Article  MATH  MathSciNet  Google Scholar 

  25. Nakayama, T.: On some modular properties of irreducible representations of the symmetric group. Jpn J. Math. 17, 165–184, 411–423 (1940)

    Google Scholar 

  26. Okounkov A. and Olshanski G.I. (1998). Shifted Schur functions. St. Petersburg Math. J. 9: 239–300

    MathSciNet  Google Scholar 

  27. Ram A. (1991). A Frobenius formula for the characters of the Hecke algebras. Invent. Math. 106: 461–488

    Article  MATH  MathSciNet  Google Scholar 

  28. Ram A. and Remmel J.B. (1997). Applications of the Frobenius formulas for the characters of the symmetric group and the Hecke algebras of type A. J. Algebr. Comb. 6: 59–87

    Article  MATH  MathSciNet  Google Scholar 

  29. Suzuki M. (1987). The values of irreducible characters of the symmetric group. Am. Math. Soc. Proc. Symp. Pure Math. 47: 317–319

    Google Scholar 

  30. Vershik A.M. and Kerov S.V. (1981). Asymptotic theory of characters of symmetric groups. Funct. Anal. Appl. 15: 246–255

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Michel Lassalle.

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Lassalle, M. An explicit formula for the characters of the symmetric group. Math. Ann. 340, 383–405 (2008). https://doi.org/10.1007/s00208-007-0156-5

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