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Bergvelt, Maarten Kedem, Rinat Di Francesco, Philippe Nevins, Thomas Addabbo, Darlayne 2017-09-29T16:39:21Z 2017-09-29T16:39:21Z 2019-09-30T09:15:08Z 2017-07-10 2017-08 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). Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2019-08-01 The student, Darlayne Addabbo, accepted the attached license on 2017-07-07 at 12:16. The student, Darlayne Addabbo, submitted this Dissertation for approval on 2017-07-07 at 12:25. This Dissertation was approved for publication on 2017-07-10 at 09:53. DSpace SAF Submission Ingestion Package generated from Vireo submission #11335 on 2017-09-29 at 11:14:50 Made available in DSpace on 2017-09-29T16:39:21Z (GMT). No. of bitstreams: 2 ADDABBO-DISSERTATION-2017.pdf: 1313831 bytes, checksum: b4973fd8b40edad8dd3f5ee3672e0c93 (MD5) LICENSE.txt: 4213 bytes, checksum: 240fbf0fe4be17a64ae34eb4b779ad3b (MD5) Previous issue date: 2017-07-10 Embargo set by: Colleen Fallaw for item 103404 Lift date: 2019-09-29T16:39:52Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system Embargo set by: Colleen Fallaw for item 103404 Lift date: 2019-09-29T17:52:45Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system Limited Restriction Lifted for Item 103404 on 2019-09-30T09:15:08Z. application/pdf http://hdl.handle.net/2142/98257 en Copyright 2017 Darlayne Addabbo Q-systems Representation theory Integrable systems Box and ball systems Q-systems and generalizations in representation theory text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.