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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 Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Q-systems 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 Type: Text URI: http://hdl.handle.net/2142/98257 Rights Information: Copyright 2017 Darlayne Addabbo Date Available in IDEALS: 2017-09-29 Date Deposited: 2017-08