Schur Superpolynomials: Combinatorial Definition and Pieri Rule
Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit q=t=0 and q=t→∞, corresponding respectively to the Schur superpolynomials and their dual. However,...
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| Дата: | 2015 |
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| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2015
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146998 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Schur Superpolynomials: Combinatorial Definition and Pieri Rule / O. Blondeau-Fournier, P. Mathieu // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1469982025-02-10T05:35:20Z Schur Superpolynomials: Combinatorial Definition and Pieri Rule Blondeau-Fournier, O. Mathieu, P. Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit q=t=0 and q=t→∞, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schur superpolynomials. Indeed, and this is an interesting novelty of the super case, the successive insertions of rows governed by this Pieri rule do not generate the tableaux underlying the Schur superpolynomials combinatorial construction, but rather those pertaining to their dual versions. As an aside, we present various extensions of the Schur bilinear identity. We thank Luc Lapointe for useful discussions and critical comments on the manuscript. We also thank Patrick Desrosiers for his collaboration at the early stages of this project. This work is supported by NSERC and FRQNT. 2015 Article Schur Superpolynomials: Combinatorial Definition and Pieri Rule / O. Blondeau-Fournier, P. Mathieu // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 14 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05E05 DOI:10.3842/SIGMA.2015.021 https://nasplib.isofts.kiev.ua/handle/123456789/146998 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit q=t=0 and q=t→∞, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schur superpolynomials. Indeed, and this is an interesting novelty of the super case, the successive insertions of rows governed by this Pieri rule do not generate the tableaux underlying the Schur superpolynomials combinatorial construction, but rather those pertaining to their dual versions. As an aside, we present various extensions of the Schur bilinear identity. |
| format |
Article |
| author |
Blondeau-Fournier, O. Mathieu, P. |
| spellingShingle |
Blondeau-Fournier, O. Mathieu, P. Schur Superpolynomials: Combinatorial Definition and Pieri Rule Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Blondeau-Fournier, O. Mathieu, P. |
| author_sort |
Blondeau-Fournier, O. |
| title |
Schur Superpolynomials: Combinatorial Definition and Pieri Rule |
| title_short |
Schur Superpolynomials: Combinatorial Definition and Pieri Rule |
| title_full |
Schur Superpolynomials: Combinatorial Definition and Pieri Rule |
| title_fullStr |
Schur Superpolynomials: Combinatorial Definition and Pieri Rule |
| title_full_unstemmed |
Schur Superpolynomials: Combinatorial Definition and Pieri Rule |
| title_sort |
schur superpolynomials: combinatorial definition and pieri rule |
| publisher |
Інститут математики НАН України |
| publishDate |
2015 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146998 |
| citation_txt |
Schur Superpolynomials: Combinatorial Definition and Pieri Rule / O. Blondeau-Fournier, P. Mathieu // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 14 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2025-07-11T01:07:20Z |
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2025-09-17T05:26:03Z |
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1844140273998757888 |