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

TL;DR

This paper introduces generalized Schur polynomials for Lie algebras, derives formulas for tau functions of Drinfeld-Sokolov hierarchies, and provides methods to compute polynomial tau functions with concrete examples.

Contribution

It defines new Schur polynomials for Lie algebra representations, derives tau function formulas, and offers computational techniques for polynomial solutions.

Findings

Tau functions are linear combinations of generalized Schur polynomials.

Explicit polynomial tau functions are computed for low-rank Lie algebras.

Bilinear equations for the DS hierarchy are identified using these tau functions.

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 Pl\"ucker 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.