Lusztig Factorization Dynamics of the Full Kostant-Toda Lattices

TL;DR

This paper explores the dynamics of the Full Kostant-Toda Lattice through Lie-theoretic factorizations, introducing new formulations, conditions for well-posedness, and novel integrable systems.

Contribution

It extends classical Toda lattices to full Hessenberg matrices, utilizing Lusztig's coordinates for precise dynamics analysis and introduces a minimal box-ball system for Full Toda.

Findings

Derived conditions for well-posedness of dynamics

Formulated a minimal box-ball system without capacities or colorings

Extended O'Connell's ODEs to the Full Toda system

Abstract

We study extensions of the classical Toda lattices at several different space-time scales. These extensions are from the classical tridiagonal phase spaces to the phase space of full Hessenberg matrices, referred to as the Full Kostant-Toda Lattice. Our formulation makes it natural to make further Lie-theoretic generalizations to dual spaces of Borel Lie algebras. Our study brings into play factorizations of Loewner-Whitney type in terms of canonical coordinatizations due to Lusztig. Using these coordinates we formulate precise conditions for the well-posedness of the dynamics at the different space-time scales. Along the way we derive a novel, minimal box-ball system for Full Toda that doesn't involve any capacities or colorings, as well as an extension of O'Connell's ODEs to Full Toda.