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

TL;DR

This paper investigates the full Kostant-Toda hierarchy and its reductions for Lie algebras of types A, B, and G, providing explicit formulas for solutions using Schur functions and root space reductions.

Contribution

It introduces explicit polynomial solutions for the Kostant-Toda hierarchy and its reductions on specific Lie algebras using root space and Chevalley system methods.

Findings

Explicit formulas for polynomial solutions using Schur functions.

Reduction of the hierarchy to $ ext{ extlangle} extell extgreater$-banded forms.

Application to Lie algebras of types 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.