Explicit Semi-invariants and Integrals of the Full Symmetric sl(n) Toda Lattice

TL;DR

This paper introduces a new method for constructing semi-invariants and integrals of the full symmetric sl(n) Toda lattice, providing explicit formulas that simplify computations for all ranks.

Contribution

The authors develop a novel approach that yields explicit formulas for semi-invariants and integrals, overcoming computational challenges of previous methods.

Findings

Explicit formulas for semi-invariants and integrals are derived.

The new approach simplifies calculations for higher-rank Lax matrices.

The method applies to all n in the full symmetric sl(n) Toda lattice.

Abstract

We show how to construct semi-invariants and integrals of the full symmetric sl(n) Toda lattice for all n. Using the Toda equations for the Lax eigenvector matrix we prove the existence of semi-invariants which are homogeneous coordinates in the corresponding projective spaces. Then we use these semi-invariants to construct the integrals. The existence of additional integrals which constitute a full set of independent non-involutive integrals was known but the chopping and Kostant procedures have crucial computational complexities already for low-rank Lax matrices and are practically not applicable for higher ranks. Our new approach solves this problem and results in simple explicit formulae for the full set of independent semi-invariants and integrals expressed in terms of the Lax matrix and its eigenvectors, and of eigenvalue matrices for the full symmetric sl(n) Toda lattice.