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arXiv · 1709.07309

Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials

Mattia Cafasso·Ann du Crest de Villeneuve·Di Yang
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Abstract

For a simple Lie algebra $\mathfrak{g}$ and an irreducible faithful representation $\pi$ of $\mathfrak{g}$, we introduce the Schur polynomials of $(\mathfrak{g},\pi)$-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 $(\mathfrak{g},\pi)$-type with the coefficients being the Pl\"ucker 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.

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Mattia Cafasso, Ann du Crest de Villeneuve, Di Yang. 2017-09-21. Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials. https://doi.org/10.3842/sigma.2018.104

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