
SIGMA 12 (2016), 033, 27 pages arXiv:1511.06721
https://doi.org/10.3842/SIGMA.2016.033
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications
Orthogonality Measure on the Torus for VectorValued Jack Polynomials
Charles F. Dunkl
Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 229044137, USA
Received November 26, 2015, in final form March 23, 2016; Published online March 27, 2016
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, selfadjoint with respect to certain Hermitian forms. These polynomials were studied by the author and J.G. Luque using a YangBaxter graph technique. This paper constructs a matrixvalued measure on the $N$torus for which the polynomials are mutually orthogonal. The construction uses Fourier analysis techniques. Recursion relations for the FourierStieltjes 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 firstorder system of differential equations.
Key words:
nonsymmetric Jack polynomials; FourierStieltjes coefficients; matrixvalued measure; symmetric group modules.
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