Vector-Valued Jack Polynomials from Scratch
Vector-valued Jack polynomials associated to the symmetric group S_N are polynomials with multiplicities in an irreducible module of S_N and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
National Academy of Science of Ukraine
2011-03-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.3842/SIGMA.2011.026 |
| _version_ | 1811239690430316544 |
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| author | Jean-Gabriel Luque Charles F. Dunkl |
| author_facet | Jean-Gabriel Luque Charles F. Dunkl |
| author_sort | Jean-Gabriel Luque |
| collection | DOAJ |
| description | Vector-valued Jack polynomials associated to the symmetric group S_N are polynomials with multiplicities in an irreducible module of 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. |
| first_indexed | 2024-04-12T13:04:25Z |
| format | Article |
| id | doaj.art-1e0ff3777d1a4ec398d3331c0eca9b5a |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-04-12T13:04:25Z |
| publishDate | 2011-03-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-1e0ff3777d1a4ec398d3331c0eca9b5a2022-12-22T03:32:04ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592011-03-017026Vector-Valued Jack Polynomials from ScratchJean-Gabriel LuqueCharles F. DunklVector-valued Jack polynomials associated to the symmetric group S_N are polynomials with multiplicities in an irreducible module of 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.http://dx.doi.org/10.3842/SIGMA.2011.026Jack polynomialsYang-Baxter graphHecke algebra |
| spellingShingle | Jean-Gabriel Luque Charles F. Dunkl Vector-Valued Jack Polynomials from Scratch Symmetry, Integrability and Geometry: Methods and Applications Jack polynomials Yang-Baxter graph Hecke algebra |
| title | Vector-Valued Jack Polynomials from Scratch |
| title_full | Vector-Valued Jack Polynomials from Scratch |
| title_fullStr | Vector-Valued Jack Polynomials from Scratch |
| title_full_unstemmed | Vector-Valued Jack Polynomials from Scratch |
| title_short | Vector-Valued Jack Polynomials from Scratch |
| title_sort | vector valued jack polynomials from scratch |
| topic | Jack polynomials Yang-Baxter graph Hecke algebra |
| url | http://dx.doi.org/10.3842/SIGMA.2011.026 |
| work_keys_str_mv | AT jeangabrielluque vectorvaluedjackpolynomialsfromscratch AT charlesfdunkl vectorvaluedjackpolynomialsfromscratch |