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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2018
• Main Authors: Cafasso, M., Du Crest De Villeneuve, A., Yang, D.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2018
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/209853
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials / M. Cafasso, A. Du Crest De Villeneuve, D. Yang // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 41 назв. — англ.

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.