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

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 Plucker 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.