A recursion and a combinatorial formula for Jack polynomials

Abstract

Heckman and Opdam introduced a non-symmetric analogue of Jack polynomials using Cherednik operators. In this paper, we derive a simple recursion formula for these polynomials and formulas relating the symmetric Jack polynomials with the non-symmetric ones. These formulas are then implemented by a closed expression of symmetric and non-symmetric Jack polynomials in terms of certain tableaux. The main application is a proof of a conjecture of Macdonald stating certain integrality and positivity properties of Jack polynomials.

Preprint March 1996, to appear in Invent. Math., 15 pages, Plain TeX

References in corpus (1)

• Exact operator solution of the Calogero-Sutherland model

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• Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r,p,n)
• On Kerov polynomials for Jack characters
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• Rodrigues Formula for the Nonsymmetric Multivariable Hermite Polynomial
• Cumulants of Jack symmetric functions and $b$-conjecture
• Cyclic sieving, skew Macdonald polynomials and Schur positivity
• Rodrigues Formula for the Nonsymmetric Multivariable Laguerre Polynomial
• Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals
• Singular Polynomials and Modules for the Symmetric Groups
• Jack vertex operators and realization of Jack functions
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• A Limit Relation for Dunkl-Bessel Functions of Type A and B
• Shifted symmetric functions and multirectangular coordinates of Young diagrams
• Introduction to double Hecke algebras
• A new scalar product for nonsymmetric Jack polynomials
• Intertwining Operators for a Degenerate Double Affine Hecke Algebra and Multivariable Orthogonal Polynomials
• Biorthogonal Expansion of Non-Symmetric Jack Functions
• From Double Hecke algebra to Fourier transform
• Involutions of double affine Hecke algebras
• Beta distributions and Sonine integrals for Bessel functions on symmetric cones
• On the evaluation formula for Jack polynomials with prescribed symmetry
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• Singular Nonsymmetric Macdonald Polynomials and Quasistaircases
• Probabilistic operators for non-attacking tableaux and a compact formula for the symmetric Macdonald polynomials
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