Skip to main content
Log in

Gaussian Limit for Projective Characters of Large Symmetric Groups

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In 1993, S. Kerov obtained a central limit theorem for the Plancherel measure on Young diagrams. The Plancherel measure is a natural probability measure on the set of irreducible characters of the symmetric group S n. Kerov's theorem states that, as n→∞, the values of irreducible characters at simple cycles, appropriately normalized and considered as random variables, are asymptotically independent and converge to Gaussian random variables. In the present work we obtain an analog of this theorem for projective representations of the symmetric group. Bibliography: 27 titles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. F. A. Berezin, Introduction to Algebra and Analysis with Anti-Commuting Variables [in Russian], Moscow University, Moscow (1983).

    Google Scholar 

  2. A. M. Borodin, “Multiplicative central measures on the Schur graph,” Zap. Nauchn. Semin. POMI, 240, 44–52 (1997).

    Google Scholar 

  3. A. M. Vershik and S. V. Kerov “Asymptotic theory of characters of the symmetric group,” Funkts. Anal. Prilozh., 15,No. 4, 15–27 (1981).

    Google Scholar 

  4. V. N. Ivanov and S. V. Kerov, “The algebra of conjugacy classes in symmetric groups, and partial permutations,” Zap. Nauchn. Semin. POMI, 256, 95–120 (1999).

    Google Scholar 

  5. V. N. Ivanov, “Dimension of skew Young diagrams and projective characters of the infinite symmetric group,” Zap. Nauchn. Semin. POMI, 240, 115–135 (1997).

    Google Scholar 

  6. D. A. Lejtes, “Introduction to the theory of supermanifolds,” Russian Math. Surveys, 35,No. 1, 1–64 (1980).

    Google Scholar 

  7. A. Yu. Okounkov and G. I. Olshansky, “Shifted Schur functions,” Algebra Analiz, 9,No. 2, 73–146 (1997).

    Google Scholar 

  8. A. N. Sergeev, “Tensor algebra of the identity representation as a module over Lie superalgebras Gl(n, m) and Q(n),” Mat. Sb., 123,No. 3, 422–430 (1984).

    Google Scholar 

  9. A. N. Sergeev, Laplace Operators and Representations of Lie Superalgebras [in Russian], Thesis, Moscow State University, Moscow (1985).

    Google Scholar 

  10. A. N. Shirjaev, Probability [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  11. A. Borodin and G. Olshanski, “Harmonic functions on multiplicative graphs and interpolation polynomials,” Electronic J. Comb., 7 (2000).

  12. P. N. Hoffman and J. F. Humphreys, Projective Representations of the Symmetric Groups, Oxford University Press (1992).

  13. A. Hora, “Central limit theorem for the adjacency operators on the infinite symmetric group,” Comm. Math. Phys., 195, 405–416 (1998).

    Google Scholar 

  14. V. Ivanov and G. Olshanski, “Kerov's central limit theorem for the Plancherel measure on Young diagrams,” (to appear).

  15. T. Józefiak, “Characters of projective representations of symmetric groups,” Expositiones Math., 7, 193–247 (1989).

    Google Scholar 

  16. T. Józefiak, “Semisimple superalgebras,” Lect. Notes Math., 1352, 96–113 (1988).

  17. S. Kerov, G. Olshanski, and A. Vershik, “Harmonic analysis on the infinite symmetric group. A deformation of the regular representation,” C. R. Acad. Sci. Paris, Série I, 316, 773–778 (1993).

    Google Scholar 

  18. S. Kerov and G. Olshanski, “Polynomial functions on the set of Young diagrams,” C. R. Acad. Sci. Paris, Série I, 319, 121–126 (1994).

    Google Scholar 

  19. S. Kerov and A. Vershik, “The Grotendieck group of infinite symmetric group and symmetric functions (with elements of the theory of K 0-functor of AF-algebras),” in: Lie Algebras and Representation Theory, Gordon and Breach (1988), pp. 37–114.

  20. S. Kerov, “Gaussian limit for the Plancherel measure of the symmetric group,” C. R. Acad. Sci. Paris, Série I, 316, 303–308 (1993).

    Google Scholar 

  21. I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, New York (1995).

    Google Scholar 

  22. A. O. Morris, “A survey on Hall-Littlewood functions and their application to representation theory,” Lect. Notes Math., 579, 136–154 (1977).

    Google Scholar 

  23. M. L. Nazarov, “Projective representations of the infinite symmetric group,” in: Representation Theory and Dynamical Systems, A. M. Vershik (ed.), Advances in Soviet Mathematics, Amer. Math. Soc., 9, 115–130 (1992).

  24. P. Pragacz, “Algebro-geometric applications of Schur S-and Q-polynomials,” Lect. Notes Math., 1478, 130–191 (1991).

    Google Scholar 

  25. I. Schur, “Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrocheme lineare Substitionen,” J. Reine Angew. Math., 139, 155–250 (1911).

    Google Scholar 

  26. J. R. Stembridge, “Shifted tableaux and the projective representations of symmetric groups,” Advances Math., 74, 87–134 (1989).

    Google Scholar 

  27. M. Yamaguchi, “A duality of the twisted group algebra of the symmetric group and a Lie superalgebra,” J. Algebra, 222,No. 1, 301–327 (1999).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ivanov, V.N. Gaussian Limit for Projective Characters of Large Symmetric Groups. Journal of Mathematical Sciences 121, 2330–2344 (2004). https://doi.org/10.1023/B:JOTH.0000024615.07311.fe

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOTH.0000024615.07311.fe

Keywords

Navigation