On the invariants of the full symmetric Toda system

TL;DR

This paper explores geometric invariants of the full symmetric Toda system, providing a new construction of commuting vector fields on compact groups that generalizes previous invariants for special linear groups.

Contribution

It introduces a simple geometric construction of a family of commuting vector fields for the Toda system, independent of the splitness of the Cartan pair, extending prior invariants.

Findings

Constructed a family of commuting vector fields on compact groups.

Extended invariants of the Toda system beyond $SL_n$.

Provided a representation-based approach not relying on split Cartan pairs.

Abstract

In this paper we continue our study of the geometric properties of full symmetric Toda systems from \cite{CSS14,CSS17,CSS19}. Namely we describe here a simple geometric construction of a commutative family of vector fields on compact groups, that include the Toda vector field, i.e. the field, which generates the full symmetric Toda system associated with the Cartan decomposition of a semisimple Lie algebra. Our construction makes use of the representations of the semisimple algebra and does not depend on the splitness of the Cartan pair. It is very close to the family of invariants and semiinvariants of the Toda system associted with $SL_n$, introduced in \cite{CS}.