Combinatorial interpretation and positivity of Kerov's character polynomials

Abstract

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).

33 pages, 13 figures, version 3: minor modifcations

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Cited by in corpus (19)

• Gaussian fluctuations of Young diagrams and structure constants of Jack characters
• Jack polynomials and free cumulants
• Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations
• Jack polynomials and orientability generating series of maps
• Zonal polynomials via Stanley's coordinates and free cumulants
• Structure coefficients for Jack characters: approximate factorization property
• Shifted symmetric functions and multirectangular coordinates of Young diagrams
• Stanley character polynomials
• Asymptotic results for Representation Theory
• A Spin Analogue of Kerov Polynomials
• Quasi-symmetric functions as polynomial functions on Young diagrams
• Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category
• Planar algebras for the Young graph and the Khovanov Heisenberg category
• Asymptotyczna teoria reprezentacji grup permutacji (Asymptotic representation theory of symmetric groups)
• Application of graph combinatorics to rational identities of type A
• A new explicit formula for Kerov polynomials
• Cyclic inclusion-exclusion
• Bijection between trees in Stanley character formula and factorizations of a cycle
• Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures