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

Abstract

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.