A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
For each irreducible module of the symmetric group SN 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 ca...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2017 |
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Інститут математики НАН України
2017
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148638 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus / C.F. Dunkl // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 11 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862608503175118848 |
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| author | Dunkl, C.F. |
| author_facet | Dunkl, C.F. |
| citation_txt | A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus / C.F. Dunkl // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | For each irreducible module of the symmetric group SN 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 xi=xj, 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.
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| first_indexed | 2025-11-28T17:34:37Z |
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| id | nasplib_isofts_kiev_ua-123456789-148638 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-28T17:34:37Z |
| publishDate | 2017 |
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| spelling | Dunkl, C.F. 2019-02-18T16:50:12Z 2019-02-18T16:50:12Z 2017 A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus / C.F. Dunkl // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 11 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C52; 32W50; 35F35; 20C30; 42B05 DOI:10.3842/SIGMA.2017.040 https://nasplib.isofts.kiev.ua/handle/123456789/148638 For each irreducible module of the symmetric group SN 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 xi=xj, 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. Some of these results were presented at the conference “Dunkl operators, special functions and
 harmonic analysis” held at Universit¨at Paderborn, Germany, August 8–12, 2016. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus Article published earlier |
| spellingShingle | A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus Dunkl, C.F. |
| title | A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus |
| title_full | A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus |
| title_fullStr | A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus |
| title_full_unstemmed | A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus |
| title_short | A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus |
| title_sort | linear system of differential equations related to vector-valued jack polynomials on the torus |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148638 |
| work_keys_str_mv | AT dunklcf alinearsystemofdifferentialequationsrelatedtovectorvaluedjackpolynomialsonthetorus AT dunklcf linearsystemofdifferentialequationsrelatedtovectorvaluedjackpolynomialsonthetorus |