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Title:  Qsystems 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 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Qsystems
Representation theory Integrable systems Box and ball systems 
Abstract:  We study taufunctions 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 taufunctions are equal to Hankel determinants and applying the famous DesnanotJacobi 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 taufunctions and the discrete equations they satisfy for the $n>2$ cases. We generalize this work by studying taufunctions 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 taufunctions which have a simple determinantal formula and the relations they satisfy are again obtained by applying the DesnanotJacobi identity. In this case, the taufunctions satisfy $T$system relations. In the following, we will define our taufunctions 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 ultradiscretization 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 ultradiscretized 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 ultradiscretizing the $T$system (discrete finite $2$dimensional Toda molecule equation). 
Issue Date:  20170710 
Type:  Text 
URI:  http://hdl.handle.net/2142/98257 
Rights Information:  Copyright 2017 Darlayne Addabbo 
Date Available in IDEALS:  20170929 20190930 
Date Deposited:  201708 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois