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

2017/09/21 by Cafasso, Mattia, de Villeneuve, Ann du Crest, Yang, Di
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1709.07309

Abstract

For a simple Lie algebra \mathfrakg and an irreducible faithful representation π of \mathfrakg, we introduce the Schur polynomials of (\mathfrakg,π)-type. We then derive the Sato-Zhou type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of \mathfrakg-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of (\mathfrakg,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 \mathfrakg 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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