{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/97314"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/97314","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Cluster algebras and discrete integrable systems","abstract":"This dissertation presents connections between cluster algebras and discrete integrable systems, especially T-systems and their specializations/generalizations. We give connections between the T-system or the octahedron relation, and the pentagram map and its various generalizations. A solution to the T-system with quasi-periodic boundary conditions gives rise to a solution to a higher pentagram map. In order to obtain all the solutions of higher pentagram map, we define T-systems with principal coefficients from cluster algebra aspect. Combinatorial solutions of the T-systems with principal coefficients with respect to any valid initial condition are shown to be partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2016). We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems of Goncharov and Kenyon 2013. We show that all Hamiltonians, partition functions of all weighted perfect matchings with a common homology class, are invariant under a move on the weighted graph. This move coincides with a cluster mutation, analog to Y-seed mutation in dimer integrable systems. Q-systems are reductions of T-systems by forgetting one of the parameters. We construct graphs for Q-systems of type A and B and show that the Hamiltonians are conserved quantities of the systems. The conserved quantities can be written as partition functions of hard particles on a certain graph. For type A, they Poisson commute under a nondegenerate Poisson bracket.","abstract_html":"This dissertation presents connections between cluster algebras and discrete integrable systems, especially T-systems and their specializations/generalizations. We give connections between the T-system or the octahedron relation, and the pentagram map and its various generalizations. A solution to the T-system with quasi-periodic boundary conditions gives rise to a solution to a higher pentagram map. In order to obtain all the solutions of higher pentagram map, we define T-systems with principal coefficients from cluster algebra aspect. Combinatorial solutions of the T-systems with principal coefficients with respect to any valid initial condition are shown to be partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2016). We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems of Goncharov and Kenyon 2013. We show that all Hamiltonians, partition functions of all weighted perfect matchings with a common homology class, are invariant under a move on the weighted graph. This move coincides with a cluster mutation, analog to Y-seed mutation in dimer integrable systems. Q-systems are reductions of T-systems by forgetting one of the parameters. We construct graphs for Q-systems of type A and B and show that the Hamiltonians are conserved quantities of the systems. The conserved quantities can be written as partition functions of hard particles on a certain graph. For type A, they Poisson commute under a nondegenerate Poisson bracket.","abstract_has_math":false,"creators":["Vichitkunakorn, Panupong"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Kedem, Rinat","Di Francesco, Philippe","Bergvelt, Maarten","Yong, Alexander"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-08-10T19:14:47Z","date_published":"2017-08-10T19:14:47Z","updated_at":"2026-07-22T22:24:32Z","subjects":["Cluster algebras","Discrete integrable systems"],"languages":["en"],"rights":["Copyright 2017 Panupong Vichitkunakorn"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/97314","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Kedem, Rinat","Di Francesco, Philippe","Bergvelt, Maarten","Yong, Alexander"]},{"key":"dc:creator","label":"Author","values":["Vichitkunakorn, Panupong"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-08-10T19:14:47Z","2017-04-13","2017-05"]},{"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":["Cluster algebras","Discrete integrable 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 Panupong Vichitkunakorn"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/97314"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This dissertation presents connections between cluster algebras and discrete integrable systems, especially T-systems and their specializations/generalizations. We give connections between the T-system or the octahedron relation, and the pentagram map and its various generalizations. A solution to the T-system with quasi-periodic boundary conditions gives rise to a solution to a higher pentagram map. In order to obtain all the solutions of higher pentagram map, we define T-systems with principal coefficients from cluster algebra aspect. Combinatorial solutions of the T-systems with principal coefficients with respect to any valid initial condition are shown to be partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2016). We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems of Goncharov and Kenyon 2013. We show that all Hamiltonians, partition functions of all weighted perfect matchings with a common homology class, are invariant under a move on the weighted graph. This move coincides with a cluster mutation, analog to Y-seed mutation in dimer integrable systems. Q-systems are reductions of T-systems by forgetting one of the parameters. We construct graphs for Q-systems of type A and B and show that the Hamiltonians are conserved quantities of the systems. The conserved quantities can be written as partition functions of hard particles on a certain graph. For type A, they Poisson commute under a nondegenerate Poisson bracket.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-08-10 without embargo terms","The student, Panupong Vichitkunakorn, accepted the attached license on 2017-04-07 at 08:14.","The student, Panupong Vichitkunakorn, submitted this Dissertation for approval on 2017-04-07 at 08:33.","This Dissertation was approved for publication on 2017-04-13 at 12:17.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10667 on 2017-08-10 at 13:38:56","Made available in DSpace on 2017-08-10T19:14:47Z (GMT). No. of bitstreams: 3 VICHITKUNAKORN-DISSERTATION-2017.pdf: 1272043 bytes, checksum: db015bf04320f0542ec691aacbbfcafd (MD5) LICENSE.txt: 4220 bytes, checksum: 559f20a3f4d163a1d070eb45739a89f0 (MD5) PROQUEST_LICENSE.txt: 4566 bytes, checksum: b147c72a907b33fcb4c67aad8eaece67 (MD5) Previous issue date: 2017-04-13"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Cluster algebras and discrete integrable systems"]}]}],"canonical_facts":{"dc:contributor":["Kedem, Rinat","Di Francesco, Philippe","Bergvelt, Maarten","Yong, Alexander"],"dc:creator":["Vichitkunakorn, Panupong"],"dc:date":["2017-08-10T19:14:47Z","2017-04-13","2017-05"],"dc:description":["This dissertation presents connections between cluster algebras and discrete integrable systems, especially T-systems and their specializations/generalizations. We give connections between the T-system or the octahedron relation, and the pentagram map and its various generalizations. A solution to the T-system with quasi-periodic boundary conditions gives rise to a solution to a higher pentagram map. In order to obtain all the solutions of higher pentagram map, we define T-systems with principal coefficients from cluster algebra aspect. Combinatorial solutions of the T-systems with principal coefficients with respect to any valid initial condition are shown to be partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2016). We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems of Goncharov and Kenyon 2013. We show that all Hamiltonians, partition functions of all weighted perfect matchings with a common homology class, are invariant under a move on the weighted graph. This move coincides with a cluster mutation, analog to Y-seed mutation in dimer integrable systems. Q-systems are reductions of T-systems by forgetting one of the parameters. We construct graphs for Q-systems of type A and B and show that the Hamiltonians are conserved quantities of the systems. The conserved quantities can be written as partition functions of hard particles on a certain graph. For type A, they Poisson commute under a nondegenerate Poisson bracket.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-08-10 without embargo terms","The student, Panupong Vichitkunakorn, accepted the attached license on 2017-04-07 at 08:14.","The student, Panupong Vichitkunakorn, submitted this Dissertation for approval on 2017-04-07 at 08:33.","This Dissertation was approved for publication on 2017-04-13 at 12:17.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10667 on 2017-08-10 at 13:38:56","Made available in DSpace on 2017-08-10T19:14:47Z (GMT). No. of bitstreams: 3 VICHITKUNAKORN-DISSERTATION-2017.pdf: 1272043 bytes, checksum: db015bf04320f0542ec691aacbbfcafd (MD5) LICENSE.txt: 4220 bytes, checksum: 559f20a3f4d163a1d070eb45739a89f0 (MD5) PROQUEST_LICENSE.txt: 4566 bytes, checksum: b147c72a907b33fcb4c67aad8eaece67 (MD5) Previous issue date: 2017-04-13"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/97314"],"dc:language":["en"],"dc:rights":["Copyright 2017 Panupong Vichitkunakorn"],"dc:subject":["Cluster algebras","Discrete integrable systems"],"dc:title":["Cluster algebras and discrete integrable systems"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:32Z"}