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Symmetry, Integrability and Geometry: Methods and Applications, 2017, Volume 13, 040, 41 pp.
DOI: https://doi.org/10.3842/SIGMA.2017.040
(Mi sigma1240)
 

This article is cited in 2 scientific papers (total in 2 papers)

A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus

Charles F. Dunkl

Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 22904-4137, USA
Full-text PDF (569 kB) Citations (2)
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Abstract: For each irreducible module of the symmetric group $\mathcal{S}_{N}$ 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 two Hermitian forms, one called the contravariant form and the other is with respect to a matrix-valued measure on the $N$-torus. The latter is valid for the parameter lying in an interval about zero which depends on the module. The author in a previous paper [SIGMA 12 (2016), 033, 27 pages] proved the existence of the measure and that its absolutely continuous part satisfies a system of linear differential equations. In this paper the system is analyzed in detail. The $N$-torus is divided into $(N-1)!$ connected components by the hyperplanes $x_{i}=x_{j}$, $i<j$, which are the singularities of the system. The main result is that the orthogonality measure has no singular part with respect to Haar measure, and thus is given by a matrix function times Haar measure. This function is analytic on each of the connected components.
Keywords: nonsymmetric Jack polynomials; matrix-valued weight function; symmetric group modules.
Received: December 11, 2016; in final form June 2, 2017; Published online June 8, 2017
Bibliographic databases:
Document Type: Article
Language: English
Citation: Charles F. Dunkl, “A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus”, SIGMA, 13 (2017), 040, 41 pp.
Citation in format AMSBIB
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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