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oai:www.ideals.illinois.edu:2142/113186 2023-07-11 col_2142_16340 col_2142_5131 com_2142_16339 com_2142_8903 com_2142_5130

Kedem, Rinat Di Francesco, Philippe Yong, Alexander Gekhtman, Michael Lin, Mingyan Simon 2022-01-12T22:35:13Z 2022-01-12T22:35:13Z 2024-01-12T22:35:30Z 2021-07-15 2021-08 In the first part of this thesis, we construct generalized Bäcklund-Darboux transformations between two Coxeter-Toda systems on simple Lie groups from a cluster algebraic prospective, using the cluster structure on Coxeter double Bruhat cells developed by Fock-Goncharov, Gekhtman et al. and Williams, thereby generalizing the construction developed by Gekhtman et al. We will then show that these generalized Backlund-Darboux transformations preserve Hamiltonian flows generated by the restriction of the trace function of any representation of the simple Lie group. In the second part of this thesis, we develop network formulations of the Coxeter-Toda Hamiltonians for the classical Lie groups, and use these network formulations to obtain combinatorial formulas for the conserved quantities of certain Q-systems. Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2023-08-01 The student, Mingyan Lin, accepted the attached license on 2021-07-13 at 14:20. The student, Mingyan Lin, submitted this Dissertation for approval on 2021-07-13 at 14:35. This Dissertation was approved for publication on 2021-07-15 at 08:02. DSpace SAF Submission Ingestion Package generated from Vireo submission #16911 on 2022-01-12 at 12:54:54 Made available in DSpace on 2022-01-12T22:35:13Z (GMT). No. of bitstreams: 2 LIN-DISSERTATION-2021.pdf: 883118 bytes, checksum: d68bf9a220684260758b8d5aace76e10 (MD5) LICENSE.txt: 4208 bytes, checksum: bcfb48b805723baed8d61961307b03c1 (MD5) Previous issue date: 2021-07-15 Embargo set by: Seth Robbins for item 121112 Lift date: 2024-01-12T22:35:30Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system U of I Only application/pdf http://hdl.handle.net/2142/113186 en Copyright 2021 Mingyan Lin Cluster algebras Integrable systems Generalized Bäcklund-Darboux transformations for Coxeter-Toda systems on simple Lie groups text Thesis Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.