Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials

Symmetry Integrability and Geometry-Methods and Applications

article

Charles F. Dunkl¹ 

[1] Department of Mathematics, University of Virginia

关键词: nonsymmetric Jack polynomials;    Fourier–Stieltjes coef ficients;    matrix-valued measure;    symmetric group modules;   

DOI  :  10.3842/SIGMA.2016.033

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown