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

Cited by in corpus (80)

• A Combinatorial Formula for Macdonald Polynomials
• Gaussian fluctuations of Young diagrams and structure constants of Jack characters
• Jack polynomials and free cumulants
• A combinatorial formula for non-symmetric Macdonald polynomials
• Supersymmetric Calogero-Moser-Sutherland models and Jack superpolynomials
• Jack polynomials and Hilbert schemes of points on surfaces
• On polynomials interpolating between the stationary state of a O(n) model and a Q.H.E. ground state
• Orthogonality of Jack polynomials in superspace
• Jack polynomials in superspace
• Evaluation and normalization of Jack superpolynomials
• A unified construction of generalised classical polynomials associated with operators of Calogero-Sutherland type
• Lectures on Dunkl operators
• Applications of Laplace-Beltrami operator for Jack polynomials
• Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r,p,n)
• On Kerov polynomials for Jack characters
• Jack polynomials with prescribed symmetry and hole propagator of spin Calogero-Sutherland model
• Orthogonal functions generalizing Jack polynomials
• Nonsymmetric Macdonald polynomials, Demazure modules and PBW filtration
• Vector-Valued Jack Polynomials from Scratch
• Non-semisimple Macdonald polynomials
• Inverse Harish-Chandra Transform and Difference Operators
• 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
• Singular polynomials for the symmetric groups
• 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
• Braid Structure and Raising-Lowering Operator Formalism in Sutherland Model
• A generalization of Newton's identity and Macdonald functions
• Algebraic study on the $A_{N-1}$- and $B_N$-Calogero models with bosonic, fermionic and distinguishable particles
• Diagonal Coinvariants and Double Affine Hecke Algebras
• An Explicit Formula for Symmetric Polynomials Related to the Eigenfunctions of Calogero-Sutherland Models
• Semisymmetric polynomials and the invariant theory of matrix vector pairs
• Macdonald polynomials in superspace as eigenfunctions of commuting operators
• A probabilistic interpretation of the Macdonald polynomials
• Nonsymmetric Koornwinder polynomials and duality
• Multivariable Bessel polynomials related to the hyperbolic Sutherland model with external Morse potential
• On the Schur expansion of Jack polynomials
• A geometric parametrization for the virtual Euler characteristic for the moduli spaces of real and complex algebriac curves
• Some Orthogonal Polynomials in Four Variables
• From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra
• Singular Nonsymmetric Macdonald Polynomials and Quasistaircases
• Probabilistic operators for non-attacking tableaux and a compact formula for the symmetric Macdonald polynomials
• Comparing formulas for type $G{L}_n$ Macdonald polynomials
• Comparing formulas for type $G{L}_n$ Macdonald polynomials: Supplement
• Symmetric functions in superspace: a compendium of results and open problems (including a SageMath worksheet)
• Jack polynomials and the coinvariant ring of $G(r,p,n)$
• An orthogonality relation for multivariable Bessel polynomials
• Nonsymmetric Macdonald polynomials and Demazure characters
• A product formula for certain Littlewood-Richardson coefficients for Jack and Macdonald polynomials
• Twisted Cherednik spectrum as a $q,t$-deformation
• A combinatorial formula for certain binomial coefficients for Jack polynomials
• Double Affine Hecke Algebras and Difference Fourier Transforms
• Spectral Theory of the Nazarov-Sklyanin Lax Operator
• Generating twisted Cherednik eigenfunctions
• Limits for circular Jacobi beta-ensembles
• LLT cumulants and graph coloring
• Irreducible modules for the degenerate double affine Hecke algebra of type $A$ as submodules of Verma modules
• A new formula for weight multiplicities and characters
• Multiplication of polynomials on Hermitian symmetric spaces and Littlewood-Richardson coefficients
• Hook-lengths and Pairs of Compositions
• The N=2 supersymmetric Calogero-Sutherland model and its eigenfunctions
• Odd Dunkl Operators and nilHecke Algebras
• Intertwining Operators of Double Affine Hecke Algebras
• A non-symmetric Yang-Baxter Algebra for the Quantum Nonlinear Schrödinger Model
• A non-symmetric Yang-Baxter algebra for the quantum nonlinear Schrödinger model (PhD Thesis)
• A positivity conjecture for Jack polynomials
• Jack polynomials with prescribed symmetry and some of their clustering properties