Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 104, 17 pages      arXiv:1709.07309

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

Mattia Cafasso a, Ann du Crest de Villeneuve a and Di Yang bc
a) LAREMA, Université d'Angers, 2 boulevard Lavoisier, Angers 49000, France
b) Max Planck Institute for Mathematics, Vivatsgasse 7, Bonn 53111, Germany
c) School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China

Received April 28, 2018, in final form September 19, 2018; Published online September 27, 2018

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 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.

Key words: Drinfeld-Sokolov hierarchy; tau function; generalized Schur polynomials.

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