Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials

TL;DR

This paper develops a matrix-valued measure on the N-torus to establish orthogonality of vector-valued nonsymmetric Jack polynomials, using Fourier analysis and recursion relations, and explores conditions where the measure construction fails.

Contribution

It introduces a new orthogonality measure for vector-valued Jack polynomials on the torus, advancing understanding of their spectral properties and parameter dependencies.

Findings

Constructed a matrix-valued measure on the N-torus for orthogonality.

Derived recursion relations for Fourier-Stieltjes coefficients.

Identified parameter values where the measure construction fails.

Abstract

For each irreducible module of the symmetric group on $N$ objects there is a set of parametrized nonsymmetric Jack polynomials in $N$ variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to certain Hermitian forms. These polynomials were studied by the author and J.-G. Luque using a Yang-Baxter graph technique. This paper constructs a matrix-valued measure on the $N$-torus for which the polynomials are mutually orthogonal. The construction uses Fourier analysis techniques. Recursion relations for the Fourier-Stieltjes coefficients of the measure are established, and used to identify parameter values for which the construction fails. It is shown that the absolutely continuous part of the measure satisfies a first-order system of differential equations.