The Kostant-Toda Lattice, Combinatorial Algorithms and Ultradiscrete Dynamics

Abstract

We study the relationship between the algorithm underlying the Robinson-Schensted-Knuth correspondence (Schensted insertion) and the Toda lattice, exploring this in the settings of discrete-time, ultradiscrete, and continuous-time dynamical systems. Starting with the work of Noumi and Yamada and their observation of a similarity between Hirota's discrete-time Toda lattice and Kirillov's geometric lifting of the RSK (geometric RSK) equations for Schensted insertion, we derive solutions to the former in its unbounded setting and provide an explicit embedding of geometric RSK in the discrete-time Toda lattice. Mimicking the ultradiscretisation of the discrete-time Toda lattice to the soliton cellular automaton, the box-ball system, we produce an extension of the classical box-ball system for Schensted insertion, which we call the ghost-box-ball system. We study this new cellular automaton in relation to Schensted insertion, demonstrating their equivalence, both on their respective coordinatisation and also on the algorithmic level. O'Connell et al. demonstrate an impressive treatment of the relation between a continuous version of geometric RSK and the Toda lattice. Through the introduction of dressing transformations and Painleve analysis, we reformulate some of these connections in a more integrable systems theoretic way. In this continuous setting, we also see the general Toda flows arise and present results on the Poisson geometry of the full Kostant-Toda lattice to lay the foundation for future probing of these exciting connections between algorithms, combinatorics and dynamical systems theory.