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Dual combinatorics of zonal polynomials

Abstract : In this paper we establish a new combinatorial formula for zonal polynomials in terms of power-sums. The proof relies on the sign-reversing involution principle. We deduce from it formulas for zonal characters, which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. These formulas are analogs of recent developments on irreducible character values of symmetric groups. The existence of such formulas could have been predicted from the work of M. Lassalle who formulated two positivity conjectures for Jack characters, which we prove in the special case of zonal polynomials.
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Valentin Féray, Piotr Sniady. Dual combinatorics of zonal polynomials. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.317-328, ⟨10.46298/dmtcs.2913⟩. ⟨hal-01215114⟩

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