[Paper Review] A Rodrigues formula for the Jack polynomials and the Macdonald-Stanley conjecture

Luc Lapointe, Luc Vinet|ArXiv.org|Sep 5, 1995

Advanced Combinatorial MathematicsMathematics3 references18 citations

TL;DR

This paper presents a Rodrigues-type formula for Jack polynomials using creation operators derived from Dunkl operators, providing a constructive method to generate these polynomials. The formula proves that the expansion coefficients of Jack polynomials in the monomial basis are polynomials in the parameter α with integer coefficients, confirming a weak form of the Macdonald-Stanley conjecture.

ABSTRACT

A formula of Rodrigues-type for the Jack polynomials is presented. It is seen to imply a weak form of a conjecture of Macdonald and Stanley.

Motivation & Objective

• To establish a Rodrigues-type formula for Jack polynomials using creation operators built from Dunkl operators.
• To prove that the expansion coefficients $ v_{ ho au}(α) $ of Jack polynomials in the monomial basis are polynomials in $ \alpha $ with integer coefficients.
• To provide a constructive approach to generating Jack polynomials as eigenfunctions of the Calogero-Sutherland Hamiltonian.
• To support the broader conjecture of Macdonald and Stanley on the integrality and nonnegativity of normalized coefficients in Jack polynomial expansions.

Proposed method

• Construction of creation operators $ B_i^+ $ via symmetric combinations of Dunkl operators $ D_j $, which act on symmetric functions.
• Definition of operators $ D_J = \prod_{j \in J} (D_{j_\kappa} + \kappa) $ for subsets $ J \subset \{1,\dots,n\} $ of size $ \ell $, ensuring symmetry under permutation of variables.
• Formulation of the Rodrigues formula: $ J_\lambda(x;\alpha) = (B_n^+)^{\lambda_n} (B_{n-1}^+)^{\lambda_{n-1}-\lambda_n} \cdots (B_1^+)^{\lambda_1 - \lambda_2} \cdot 1 $, generating Jack polynomials recursively.
• Verification that the action of creation operators preserves symmetry and homogeneity, and that the resulting functions satisfy the eigenvalue equation for the Calogero-Sutherland Hamiltonian $ H(\alpha) $.
• Use of commutation relations among Dunkl operators to ensure that $ \operatorname{Res}^J D_J $ is symmetric in the variables indexed by $ J $, enabling consistent action on symmetric functions.
• Recursive proof that matrix elements of $ B_i^+ $ in the monomial basis are polynomials in $ \alpha $ with integer coefficients, relying on the action of $ D_j $ on monomials.

Experimental results

Research questions

• RQ1 Can a Rodrigues-type formula be constructed for Jack polynomials using creation operators derived from Dunkl operators?
• RQ2 Do the expansion coefficients $ v_{\lambda\mu}(\alpha) $ of Jack polynomials in the monomial basis lie in $ \mathbb{Z}[\alpha] $, i.e., are they polynomials with integer coefficients?
• RQ3 Does the proposed formula imply a weak form of the Macdonald-Stanley conjecture on the integrality of normalized coefficients?
• RQ4 How can the Jack polynomials be systematically generated as eigenfunctions of the Calogero-Sutherland Hamiltonian?
• RQ5 What is the algebraic structure underlying the wave functions of the Calogero-Sutherland model, and how can it be captured via such a formula?

Key findings

• The Rodrigues formula $ J_\lambda(x;\alpha) = (B_n^+)^{\lambda_n} \cdots (B_1^+)^{\lambda_1 - \lambda_2} \cdot 1 $ provides a constructive, recursive method to generate Jack polynomials from the constant 1.
• The action of the creation operators $ B_i^+ $ on symmetric functions preserves symmetry and homogeneity, ensuring the output is a symmetric, homogeneous polynomial of degree $ N = |\lambda| $.
• The formula implies that the coefficients $ v_{\lambda\mu}(\alpha) $ in the monomial expansion of $ J_\lambda $ are polynomials in $ \alpha $ with integer coefficients, confirming a weak form of the Macdonald-Stanley conjecture.
• The eigenvalue equation $ H(\alpha)J_\lambda = \varepsilon_\lambda(\alpha)J_\lambda $ is satisfied by the Rodrigues construction, with $ \varepsilon_\lambda(\alpha) = \sum_{j=1}^n \left[ \alpha\lambda_j^2 + (n+1-2j)\lambda_j \right] $, confirming the correct spectrum.
• Matrix elements of $ B_i^+ $ in the monomial basis are shown to be polynomials in $ \alpha $ with integer coefficients, due to the action of $ D_j $ on monomials yielding such expressions.
• The result provides a foundational tool for further exploration of the algebraic structure of the Calogero-Sutherland model and extensions to Macdonald polynomials and other root systems.