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.

Tau Functions as Widom Constants

“…As a second example, we will show how to apply our definition of tau function to the study of integrable hierarchies. The results outlined in this section are not new (see [Caf,CW1,CW2,CDD]); the aim is to describe them in a way that makes the comparison with the case of isomonodromic deformations more transparent. To start with, consider a differential operator of fixed degree N…”

“…where E + is the set of the (positive) exponents of the Kac-Moody algebra g[z, z −1 ] ⊕ Cc, and Λ j , j ∈ E + is (half of ) the Heisenberg sub-algebra associated to an arbitrary gradation of the algebra. Polynomial and topological solutions of these hierarchies had been treated, using this formalism, in [CDD,CW2]. It would be interesting (but technically involved, because of the size of matrix representations) to study algebro-geometric solutions associated to arbitrary Drinfeld-Sokolov hierarchies.…”

“…On the isospectral side, the coincidence between the Widom constant and the Sato-Segal-Wilson tau function had been established, for the so-called Gelfand-Dickey hierarchies, by one of the authors in [Caf], and successively extended to the Drinfeld-Sokolov hierarchies associated to an arbitrary Kac-Moody algebra in [CW1]. Based on this identification, effective computations had been carried out in [CW2,CDD] for topological and polynomial tau functions.…”