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An explicit form for Kerov's character polynomials


Authors: I. P. Goulden and A. Rattan
Journal: Trans. Amer. Math. Soc. 359 (2007), 3669-3685
MSC (2000): Primary 05E10; Secondary 05A15, 20C30
Published electronically: February 23, 2007
MathSciNet review: 2302511
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Abstract | References | Similar Articles | Additional Information

Abstract: Kerov considered the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as a polynomial in free cumulants. Biane has proved that this polynomial has integer coefficients, and made various conjectures. Recently, Sniady has proved Biane's conjectured explicit form for the first family of nontrivial terms in this polynomial. In this paper, we give an explicit expression for all terms in Kerov's character polynomials. Our method is through Lagrange inversion.


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  • 1. Philippe Biane, Representations of symmetric groups and free probability, Adv. Math. 138 (1998), no. 1, 126–181. MR 1644993, 10.1006/aima.1998.1745
  • 2. Philippe Biane, Free cumulants and representations of large symmetric groups, XIIIth International Congress on Mathematical Physics (London, 2000), Int. Press, Boston, MA, 2001, pp. 321–326. MR 1883322
  • 3. Philippe Biane, Characters of symmetric groups and free cumulants, Asymptotic combinatorics with applications to mathematical physics (St. Petersburg, 2001) Lecture Notes in Math., vol. 1815, Springer, Berlin, 2003, pp. 185–200. MR 2009840, 10.1007/3-540-44890-X_8
  • 4. Sylvie Corteel, Alain Goupil, and Gilles Schaeffer, Content evaluation and class symmetric functions, Adv. Math. 188 (2004), no. 2, 315–336. MR 2087230, 10.1016/j.aim.2003.09.010
  • 5. Avital Frumkin, Gordon James, and Yuval Roichman, On trees and characters, J. Algebraic Combin. 17 (2003), no. 3, 323–334. MR 2001674, 10.1023/A:1025052922664
  • 6. Ian P. Goulden and David M. Jackson, Combinatorial enumeration, Dover Publications, Inc., Mineola, NY, 2004. With a foreword by Gian-Carlo Rota; Reprint of the 1983 original. MR 2079788
  • 7. V. Ivanov and G. Olshanski, Kerov's central limit theorem for the Plancherel measure on Young diagrams, Symmetric functions 2001: Surveys of developments and perspectives, S. Fomin (Ed.), NATO Science series II. Mathematics, Physics and Chemistry 74 (2002), 93-151.
  • 8. Jacob Katriel, Explicit expressions for the central characters of the symmetric group, Discrete Appl. Math. 67 (1996), no. 1-3, 149–156. MR 1393301, 10.1016/0166-218X(95)00016-K
  • 9. Serguei Kerov, Gaussian limit for the Plancherel measure of the symmetric group, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 4, 303–308 (English, with English and French summaries). MR 1204294
  • 10. I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
  • 11. Piotr Śniady, Asymptotics of characters of symmetric groups, genus expansion and free probability, Discrete Math. 306 (2006), no. 7, 624–665. MR 2215589, 10.1016/j.disc.2006.02.004
  • 12. R.P. Stanley, Kerov's character polynomial and irreducible symmetric group characters of rectangular shape, Transparencies from a talk at CMS meeting, Quebec City, 2002.

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Additional Information

I. P. Goulden
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: ipgoulden@math.uwaterloo.ca

A. Rattan
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Address at time of publication: Department of Applied Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: arattan@math.mit.ca

DOI: https://doi.org/10.1090/S0002-9947-07-04311-5
Received by editor(s): April 20, 2005
Published electronically: February 23, 2007
Article copyright: © Copyright 2007 American Mathematical Society