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Title:Q-systems and generalizations in representation theory
Author(s):Addabbo, Darlayne
Director of Research:Bergvelt, Maarten
Doctoral Committee Chair(s):Kedem, Rinat
Doctoral Committee Member(s):Di Francesco, Philippe; Nevins, Thomas
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Representation theory
Integrable systems
Box and ball systems
Abstract:We study tau-functions given as matrix elements for the action of loop groups, $\widehat{GL_n}$ on $n$-component fermionic Fock space. In the simplest case, $n=2$, the tau-functions are equal to Hankel determinants and applying the famous Desnanot-Jacobi identity, one can see that they satisfy a $Q$-system. Since $Q$-systems are of interest in many areas of mathematics, it is interesting to study tau-functions and the discrete equations they satisfy for the $n>2$ cases. We generalize this work by studying tau-functions equal to matrix elements for the action of infinite matrix groups, denoted $\widehat{GL}_{\infty}^{(n)}$ on $n$-component fermionic Fock space. The $n=2$ case, similarly to the $\widehat{GL_2}$ situation, gives tau-functions which have a simple determinantal formula and the relations they satisfy are again obtained by applying the Desnanot-Jacobi identity. In this case, the tau-functions satisfy $T$-system relations. In the following, we will define our tau-functions and explain how to compute them and then present multiple ways of deriving the relations that they satisfy, which is much more complicated in the $n>2$ cases. The method of ultra-discretization provides a way to obtain from discrete integrable equations, combinatorial models that maintain the essential properties of the original equations. With some extra initial conditions, the $Q$-system for our $\widehat{GL_2}$ case is also known as the discrete finite $1$-dimensional Toda molecule equation. It is known that this can be ultra-discretized to obtain the famous Box and Ball system. In the final chapter of this thesis, we present a new generalization of the Box and Ball system obtained by ultra-discretizing the $T$-system (discrete finite $2$-dimensional Toda molecule equation).
Issue Date:2017-07-10
Rights Information:Copyright 2017 Darlayne Addabbo
Date Available in IDEALS:2017-09-29
Date Deposited:2017-08

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