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Home > Vol 29, No 4 (2009)

Valentin Féray

Université Paris-Est The Gaspard-Monge Institut of Electronique and Computer Science 77454 Marne-la-Vallée Cedex 2 France

Kerov's polynomials give irreducible character values in terms of the free cumulants of the associated Young diagram. We prove in this article a positivity result on their coefficients, which extends a conjecture of S. Kerov. Our method, through decomposition of maps, gives a description of the coefficients of the k-th Kerov's polynomial using permutations in S( k). We also obtain explicit formulas or combinatorial interpretations for some coefficients. In particular, we are able to compute the subdominant term for character values on any fixed permutation (it was known for cycles).

1. Biane, P.: Some properties of crossings and partitions. Discrete Math. 175(1-3), 41-53 (1997)
2. Biane, P.: Representations of symmetric groups and free probability. Advances in Mathematics 138, 126-181 (1998)
3. Biane, P.: Characters of symmetric groups and free cumulants. In: Vershik, A. (ed.) Asymptotic Combinatorics with Applications to Mathematical Physics. Lecture Notes in Mathematics, vol. 1815, pp. 185-200. Springer, Berlin (2003)
4. Biane, P.: On the formula of Goulden and Rattan for Kerov polynomials. Séminaire Lotharingien de Combinatoire 55, B55d (2005/06)
5. Féray, V.: Proof of Stanley's conjecture about irreducible character values of the symmetric group. Annals of combinatorics (2008, to appear).
6. Féray, V., Śniady, P.: Asymptotics of characters of symmetric groups related to Stanley-Féray formula. (2007)
7. Goulden, I.P., Jackson, D.M.: The combinatorial relationship between trees, cacti, and certain connection coefficients for the symmetric group. European J. Combinatorics 13, 357-365 (1992)
8. Goulden, I.P., Rattan, A.: An explicit form for Kerov's character polynomials. Trans. Am. Math. Soc. 359, 3669-3685 (2007)
9. Kerov, S.: The asymptotics of interlacing sequences and the growth of continual Young diagrams. Journal of Mathematical Science 80(3), 1760-1767 (1996)
10. Lassalle, M.: Two positivity conjectures for Kerov polynomials. (2007)
11. Macdonald, I.G.: Symmetric functions and Hall polynomials. Oxford Univ. Press, Oxford (1979)
12. Rattan, A.: Stanley's character polynomials and colored factorizations in the symmetric group. (2006)
13. Rattan, A., Śniady, P.: Upper bounds on the characters of the symmetric group for balanced Young diagram and a generalized Frobenius formula. (2006)
14. Śniady, P.: Gaussian fluctuations of characters of symmetric groups and of Young diagrams. Probab. Theory Related Fields 136(2), 263-297 (2006)
15. Śniady, P.: Asymptotics of characters of symmetric groups, genus expansion and free probability.