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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Biane P. (1998). Representations of symmetric groups and free probability. Adv. Math. 138: 126–181
Biane, P.: On the formula of Goulden and Rattan for Kerov polynomials. Sém. Lothar. Combin., 55, article B55d (2006)
Corteel S., Goupil A. and Schaeffer G. (2004). Content evaluation and class symmetric functions. Adv. Math. 188: 315–336
Désarménien, J.: Une généralisation des caractères du groupe symétrique. Unpublished note (1996)
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
Garsia, A.: Young seminormal representation, Murphy elements and content evaluations. Lecture notes (march 2003), available at http://www.math.ucsd.edu/~garsia/recentpapers/
Goldschmidt, D.M.: Group characters, symmetric functions and the Hecke algebra. University Lecture Series 4. Am. Math. Soc., Providence (1991)
Goulden, I.P., Rattan, A.: An explicit form for Kerov’s character polynomials. Trans. Am. Math. Soc. 359, 3669–3685 (2007)
Ingram R.E. (1950). Some characters of the symmetric group. Proc. Am. Math. Soc. 1: 358–369
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)
Jucys A.A. (1974). Symmetric polynomials and the center of the symmetric group ring. Rep. Math. Phys. 5: 107–112
Katriel J. (1996). Explicit expressions for the central characters of the symmetric group. Discrete Appl. Math. 67: 149–156
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
Lascoux, A.: Notes on interpolation in one and several variables. Available at http://igm.univ-mlv.fr/~al/
Lascoux A. and Thibon J.-Y. (2001). Vertex operators and the class algebras of symmetric groups. Zapiski Nauchnyh Seminarov POMI 283: 156–177
Lassalle M. (1998). Some combinatorial conjectures for Jack polynomials. Ann. Comb. 2: 61–83
Lassalle M. (2001). Une q-spécialisation pour les fonctions symétriques monomiales. Adv. Math. 162: 217–242
Lassalle M. (2002). A new family of positive integers. Ann. Comb. 6: 399–405
Lassalle M. (2004). Jack polynomials and some identities for partitions. Trans. Am. Math. Soc. 356: 3455–3476
Lassalle M. (2005). Explicitation of characters of the symmetric group. C. R. Acad. Sci. Paris Sér. I 341: 529–534
Lassalle, M.: Available at http://igm.univ-mlv.fr/~lassalle/char.html
Macdonald I.G. (1995). Symmetric functions and Hall polynomials, 2nd edn. Clarendon Press, Oxford
Murnaghan F.D. (1937). On the representations of the symmetric group. Am. J. Math. 59: 739–753
Murphy G.E. (1981). A new construction of Young’s seminormal representation of the symmetric group. J. Algebra 69: 287–291
Nakayama, T.: On some modular properties of irreducible representations of the symmetric group. Jpn J. Math. 17, 165–184, 411–423 (1940)
Okounkov A. and Olshanski G.I. (1998). Shifted Schur functions. St. Petersburg Math. J. 9: 239–300
Ram A. (1991). A Frobenius formula for the characters of the Hecke algebras. Invent. Math. 106: 461–488
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
Suzuki M. (1987). The values of irreducible characters of the symmetric group. Am. Math. Soc. Proc. Symp. Pure Math. 47: 317–319
Vershik A.M. and Kerov S.V. (1981). Asymptotic theory of characters of symmetric groups. Funct. Anal. Appl. 15: 246–255
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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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DOI: https://doi.org/10.1007/s00208-007-0156-5