# Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials

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.
Domain :

https://hal.archives-ouvertes.fr/hal-01676243
Contributor : Mattia Cafasso <>
Submitted on : Friday, January 5, 2018 - 12:31:29 PM
Last modification on : Thursday, April 30, 2020 - 1:22:02 PM

### Identifiers

• HAL Id : hal-01676243, version 1
• ARXIV : 1709.07309

### Citation

Mattia Cafasso, Ann Du Crest de Villeneuve, Di Yang. Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials. 2018. ⟨hal-01676243⟩

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