Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials
For a simple Lie algebra g and an irreducible faithful representation π of g, we introduce the Schur polynomials of (g,π)-type. We then derive the Sato-Zhou-type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of g-type. Namely, we show that the tau functions are linear combinations...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2018 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2018
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/209853 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials / M. Cafasso, A. Du Crest De Villeneuve, D. Yang // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 41 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862530180576182272 |
|---|---|
| author | Cafasso, M. Du Crest De Villeneuve, A. Yang, D. |
| author_facet | Cafasso, M. Du Crest De Villeneuve, A. Yang, D. |
| citation_txt | Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials / M. Cafasso, A. Du Crest De Villeneuve, D. Yang // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 41 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | For a simple Lie algebra g and an irreducible faithful representation π of g, we introduce the Schur polynomials of (g,π)-type. We then derive the Sato-Zhou-type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of g-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of (g,π)-type with the coefficients being the Plücker coordinates. As an application, we provide a way of computing polynomial tau functions for the DS hierarchy. For g of low rank, we give several examples of polynomial tau functions and use them to detect bilinear equations for the DS hierarchy.
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| first_indexed | 2025-12-02T18:54:12Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-209853 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-02T18:54:12Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cafasso, M. Du Crest De Villeneuve, A. Yang, D. 2025-11-27T14:55:36Z 2018 Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials / M. Cafasso, A. Du Crest De Villeneuve, D. Yang // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 41 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K10; 17B80 arXiv: 1709.07309 https://nasplib.isofts.kiev.ua/handle/123456789/209853 https://doi.org/10.3842/SIGMA.2018.104 For a simple Lie algebra g and an irreducible faithful representation π of g, we introduce the Schur polynomials of (g,π)-type. We then derive the Sato-Zhou-type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of g-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of (g,π)-type with the coefficients being the Plücker coordinates. As an application, we provide a way of computing polynomial tau functions for the DS hierarchy. For g of low rank, we give several examples of polynomial tau functions and use them to detect bilinear equations for the DS hierarchy. We would like to thank Ferenc Balogh, Marco Bertola, Boris Dubrovin, John Harnad, Leonardo Patimo, Daniele Valeri, Chao-Zhong Wu, and Jian Zhou for helpful discussions. D.Y. is grateful to Youjin Zhang and Boris Dubrovin for their advice and to Victor Kac for helpful suggestions. We thank the anonymous referees for constructive comments. Part of our work was done at SISSA; we acknowledge SISSA for excellent working conditions and generous support. A.D. and M.C. thank the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment. Part of the work of D.Y. was done during his visits to LAREMA; he acknowledges the support of LAREMA and warm hospitality. A.D. and M.C. acknowledge the support of the project IPaDEGAN (H2020-MSCA-RISE-2017), Grant No. 778010. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials Article published earlier |
| spellingShingle | Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials Cafasso, M. Du Crest De Villeneuve, A. Yang, D. |
| title | Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials |
| title_full | Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials |
| title_fullStr | Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials |
| title_full_unstemmed | Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials |
| title_short | Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials |
| title_sort | drinfeld-sokolov hierarchies, tau functions, and generalized schur polynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209853 |
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