Vector-Valued Jack Polynomials from Scratch

Abstract

Vector-valued Jack polynomials associated to the symmetric group ${\mathfrak{S}}_N$ are polynomials with multiplicities in an irreducible module of ${\mathfrak{S}}_N$ and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups $G(r,p,N)$ and studied by one of the authors (C. Dunkl) in the specialization $r=p=1$ (i.e. for the symmetric group). By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.

References in corpus (5)

• A recursion and a combinatorial formula for Jack polynomials
• Clustering Properties and Model Wavefunctions for Non-Abelian Fractional Quantum Hall Quasielectrons
• Jack polynomial fractional quantum Hall states and their generalizations
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Cited by in corpus (8)

• Vector valued Macdonald polynomials
• Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials
• The Smallest Singular Values and Vector-Valued Jack Polynomials
• Singular Nonsymmetric Macdonald Polynomials and Quasistaircases
• Vector-valued Jack Polynomials and Wavefunctions on the Torus
• A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
• Some Singular Vector-valued Jack and Macdonald Polynomials
• A Superpolynomial Version of Nonsymmetric Jack Polynomials