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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 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.