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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2017
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1709.07309 |
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| _version_ | 1866909755742617600 |
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| author | Cafasso, Mattia de Villeneuve, Ann du Crest Yang, Di |
| author_facet | Cafasso, Mattia de Villeneuve, Ann du Crest Yang, Di |
| contents | For a simple Lie algebra $\mathfrak{g}$ and an irreducible faithful representation $π$ of $\mathfrak{g}$, we introduce the Schur polynomials of $(\mathfrak{g},π)$-type. We then derive the Sato-Zhou type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of $\mathfrak{g}$-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of $(\mathfrak{g},π)$-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 $\mathfrak{g}$ of low rank, we give several examples of polynomial tau functions, and use them to detect bilinear equations for the DS hierarchy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1709_07309 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials Cafasso, Mattia de Villeneuve, Ann du Crest Yang, Di Mathematical Physics Exactly Solvable and Integrable Systems For a simple Lie algebra $\mathfrak{g}$ and an irreducible faithful representation $π$ of $\mathfrak{g}$, we introduce the Schur polynomials of $(\mathfrak{g},π)$-type. We then derive the Sato-Zhou type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of $\mathfrak{g}$-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of $(\mathfrak{g},π)$-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 $\mathfrak{g}$ of low rank, we give several examples of polynomial tau functions, and use them to detect bilinear equations for the DS hierarchy. |
| title | Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials |
| topic | Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/1709.07309 |