On the full Kostant-Toda hierarchy and its $\ell$-banded reductions for the Lie algebras of type $A, B$ and $G$

Abstract

This paper concerns the solutions of the full Kostant-Toda (f-KT) hierarchy in the Hessenberg form and their reductions to the $\ell$-banded Kostant-Toda ($\ell$-KT) hierarchy. We also study the f-KT hierarchy and the corresponding $\ell$-KT hierarchy on simple Lie algebras of type $A, B$ and $G$ based on root space reductions with proper Chevalley systems. Explicit formulas of the polynomial solutions for the $\tau$-functions are also given in terms of the Schur functions and Schur's $Q$-functions.