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    The Kostant-Toda Lattice, Combinatorial Algorithms and Ultradiscrete Dynamics

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    Author
    Ramalheira-Tsu, Jonathan
    Issue Date
    2020
    Keywords
    Box-Ball System
    Combinatorics
    Dynamical Systems
    Integrable Systems
    Robinson-Schensted-Knuth Correspondence
    The Toda Lattice
    Advisor
    Ercolani, Nicholas M.
    
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    Publisher
    The University of Arizona.
    Rights
    Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction, presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
    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.
    Type
    text
    Electronic Dissertation
    Degree Name
    Ph.D.
    Degree Level
    doctoral
    Degree Program
    Graduate College
    Mathematics
    Degree Grantor
    University of Arizona
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