{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/98257"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/98257","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Q-systems and generalizations in representation theory","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).","abstract_html":"We study tau-functions given as matrix elements for the action of loop groups, <span class=\"etd-inline-math\">\\widehat{GL<sub>n</sub>}</span> 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 $n&gt;2$ cases. We generalize this work by studying tau-functions equal to matrix elements for the action of infinite matrix groups, denoted <span class=\"etd-inline-math\">\\widehat{GL}<sub>\\infty</sub><sup>(n)</sup></span> on $n$-component fermionic Fock space. The $n=2$ case, similarly to the <span class=\"etd-inline-math\">\\widehat{GL<sub>2</sub>}</span> 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 $n&gt;2$ 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 <span class=\"etd-inline-math\">\\widehat{GL<sub>2</sub>}</span> 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).","abstract_has_math":true,"creators":["Addabbo, Darlayne"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Bergvelt, Maarten","Kedem, Rinat","Di Francesco, Philippe","Nevins, Thomas"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-09-29T16:39:21Z","date_published":"2017-09-29T16:39:21Z","updated_at":"2026-07-22T22:24:35Z","subjects":["Q-systems","Representation theory","Integrable systems","Box and ball systems"],"languages":["en"],"rights":["Copyright 2017 Darlayne Addabbo"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/98257","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bergvelt, Maarten","Kedem, Rinat","Di Francesco, Philippe","Nevins, Thomas"]},{"key":"dc:creator","label":"Author","values":["Addabbo, Darlayne"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-09-29T16:39:21Z","2019-09-30T09:15:08Z","2017-07-10","2017-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Q-systems","Representation theory","Integrable systems","Box and ball systems"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2017 Darlayne Addabbo"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/98257"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Q-systems and generalizations in representation theory"]}]}],"canonical_facts":{"dc:contributor":["Bergvelt, Maarten","Kedem, Rinat","Di Francesco, Philippe","Nevins, Thomas"],"dc:creator":["Addabbo, Darlayne"],"dc:date":["2017-09-29T16:39:21Z","2019-09-30T09:15:08Z","2017-07-10","2017-08"],"dc:description":["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). 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