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      A linear system of differential equations related to vector-valued Jack polynomials on the torus

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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 2016-033) 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 \(\left( N-1\right) !\) connected components by the hyperplanes \(x_{i}=x_{j}\), \(\left( i<j\right) \) 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.

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          Harmonic analysis for certain representations of graded Hecke algebras

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            Vector-Valued Jack Polynomials from Scratch

            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.
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              Author and article information

              Journal
              2016-12-05
              Article
              1612.01486
              bfe8f1aa-8545-4323-9a14-19c00b91cf3d

              http://arxiv.org/licenses/nonexclusive-distrib/1.0/

              History
              Custom metadata
              33C52, 20C30, 35F35, 46G10, 42B10
              55 pages
              math.CA

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