Lusztig Factorization Dynamics of the Full Kostant-Toda Lattices
- Resource Type
- Working Paper
- Authors
- Ercolani, Nicholas M.; Ramalheira-Tsu, Jonathan
- Source
- Mathematical Physics, Analysis and Geometry, 26:2 (2023)
- Subject
- Nonlinear Sciences - Exactly Solvable and Integrable Systems
- Language
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.