Nonsymmetric Macdonald Superpolynomials
There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algeb...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
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Інститут математики НАН України
2021
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211295 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Nonsymmetric Macdonald Superpolynomials. Charles F. Dunkl. SIGMA 17 (2021), 054, 29 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862707593111142400 |
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| author | Dunkl, Charles F. |
| author_facet | Dunkl, Charles F. |
| citation_txt | Nonsymmetric Macdonald Superpolynomials. Charles F. Dunkl. SIGMA 17 (2021), 054, 29 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper, the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases, and their properties are first derived. In terms of the standard Young tableau approach to representations, these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here ''symmetric'' means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, as well as some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials, there is a factorization when the polynomials are evaluated at special points.
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| first_indexed | 2026-03-19T04:35:12Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211295 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T04:35:12Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
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| spelling | Dunkl, Charles F. 2025-12-29T11:04:33Z 2021 Nonsymmetric Macdonald Superpolynomials. Charles F. Dunkl. SIGMA 17 (2021), 054, 29 pages 1815-0659 2020 Mathematics Subject Classification: 33D56; 20C08; 05E05 arXiv:2011.05886 https://nasplib.isofts.kiev.ua/handle/123456789/211295 https://doi.org/10.3842/SIGMA.2021.054 There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper, the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases, and their properties are first derived. In terms of the standard Young tableau approach to representations, these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here ''symmetric'' means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, as well as some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials, there is a factorization when the polynomials are evaluated at special points. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Nonsymmetric Macdonald Superpolynomials Article published earlier |
| spellingShingle | Nonsymmetric Macdonald Superpolynomials Dunkl, Charles F. |
| title | Nonsymmetric Macdonald Superpolynomials |
| title_full | Nonsymmetric Macdonald Superpolynomials |
| title_fullStr | Nonsymmetric Macdonald Superpolynomials |
| title_full_unstemmed | Nonsymmetric Macdonald Superpolynomials |
| title_short | Nonsymmetric Macdonald Superpolynomials |
| title_sort | nonsymmetric macdonald superpolynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211295 |
| work_keys_str_mv | AT dunklcharlesf nonsymmetricmacdonaldsuperpolynomials |