Combinatorial interpretation and positivity of Kerov's character polynomials

Details Combinatorial interpretation and positivity of Kerov's character polynomials Combinatorial interpretation and positivity of Kerov's character polynomials

2007-10-31

arxiv.org/abs/0710.5885v3

J. Alg. Comb., 29 (4), pp. 473 - 507 (2009)

Mathematics

Representation Theory

33 pages, 13 figures, version 3: minor modifcations

Scientific paper

10.1007/s10801-008-0147-y

Kerov's polynomials give irreducible character values in term 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 polynomials 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).

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