{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/98093"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/98093","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Nakajima's (Q, T)-characters as quantum cluster variables","abstract":"This Dissertation was approved for publication on 2017-05-03 at 09:13.","abstract_html":"This Dissertation was approved for publication on 2017-05-03 at 09:13.","abstract_has_math":false,"creators":["Turmunkh, Bolor"],"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","Bergvelt, Maarten","Di Francesco, Philippe","Nevins, Thomas"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-09-29T16:37:43Z","date_published":"2017-09-29T16:37:43Z","updated_at":"2026-07-22T22:24:34Z","subjects":["T-system","Nakajima (q,t)-characters","Quantum cluster algebra"],"languages":["en"],"rights":["Copyright 2017 Bolor Turmunkh"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/98093","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Kedem, Rinat","Bergvelt, Maarten","Di Francesco, Philippe","Nevins, Thomas"]},{"key":"dc:creator","label":"Author","values":["Turmunkh, Bolor"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-09-29T16:37:43Z","2017-05-03","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":["T-system","Nakajima (q,t)-characters","Quantum cluster algebra"]}]},{"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 Bolor Turmunkh"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/98093"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This Dissertation was approved for publication on 2017-05-03 at 09:13.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11076 on 2017-09-29 at 11:25:37","Made available in DSpace on 2017-09-29T16:37:43Z (GMT). No. of bitstreams: 3 TURMUNKH-DISSERTATION-2017.pdf: 638677 bytes, checksum: 551735ebdf9918e64a577185ceceb0a8 (MD5) LICENSE.txt: 4211 bytes, checksum: 0873b4c992f1bceef20c89035010c3be (MD5) PROQUEST_LICENSE.txt: 4557 bytes, checksum: 13da9f2dfe922836bca534988aeec133 (MD5) Previous issue date: 2017-05-03","Nakajima introduced a t-deformation of q-characters, (q,t)-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima (q, t)-characters of Kirillov-Reshetikhin modules satisfy a t-deformed T-system. The T-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this thesis, we show that the Nakajima t-deformed T-system of type Ar forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the T-system. There are 2 main parts to our work. The bulk of the work is a combinatorial construction that proves (q, t)-characters of a certain set of Kirillov-Reshetikhin modules t-commute under Nakajima’s twisted multi- plication. We use a slightly modified version of the tableaux-sum notation for q-characters and define the notion of a block-tableau, which plays an integral role in the proof. Once t-commutativity is established, the second half of this thesis is concerned with the commutation coefficients of the given set of Kirillov-Reshetikhin modules. In particular, we show that the commutation coefficients are compatible with the cluster algebra exchange matrix and the mutation relations in the language of Berenstein-Zelevinsky.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, Bolor Turmunkh, accepted the attached license on 2017-04-25 at 20:45.","The student, Bolor Turmunkh, submitted this Dissertation for approval on 2017-04-25 at 20:50."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Nakajima's (Q, T)-characters as quantum cluster variables"]}]}],"canonical_facts":{"dc:contributor":["Kedem, Rinat","Bergvelt, Maarten","Di Francesco, Philippe","Nevins, Thomas"],"dc:creator":["Turmunkh, Bolor"],"dc:date":["2017-09-29T16:37:43Z","2017-05-03","2017-08"],"dc:description":["This Dissertation was approved for publication on 2017-05-03 at 09:13.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11076 on 2017-09-29 at 11:25:37","Made available in DSpace on 2017-09-29T16:37:43Z (GMT). No. of bitstreams: 3 TURMUNKH-DISSERTATION-2017.pdf: 638677 bytes, checksum: 551735ebdf9918e64a577185ceceb0a8 (MD5) LICENSE.txt: 4211 bytes, checksum: 0873b4c992f1bceef20c89035010c3be (MD5) PROQUEST_LICENSE.txt: 4557 bytes, checksum: 13da9f2dfe922836bca534988aeec133 (MD5) Previous issue date: 2017-05-03","Nakajima introduced a t-deformation of q-characters, (q,t)-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima (q, t)-characters of Kirillov-Reshetikhin modules satisfy a t-deformed T-system. The T-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this thesis, we show that the Nakajima t-deformed T-system of type Ar forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the T-system. There are 2 main parts to our work. The bulk of the work is a combinatorial construction that proves (q, t)-characters of a certain set of Kirillov-Reshetikhin modules t-commute under Nakajima’s twisted multi- plication. We use a slightly modified version of the tableaux-sum notation for q-characters and define the notion of a block-tableau, which plays an integral role in the proof. Once t-commutativity is established, the second half of this thesis is concerned with the commutation coefficients of the given set of Kirillov-Reshetikhin modules. In particular, we show that the commutation coefficients are compatible with the cluster algebra exchange matrix and the mutation relations in the language of Berenstein-Zelevinsky.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, Bolor Turmunkh, accepted the attached license on 2017-04-25 at 20:45.","The student, Bolor Turmunkh, submitted this Dissertation for approval on 2017-04-25 at 20:50."],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/98093"],"dc:language":["en"],"dc:rights":["Copyright 2017 Bolor Turmunkh"],"dc:subject":["T-system","Nakajima (q,t)-characters","Quantum cluster algebra"],"dc:title":["Nakajima's (Q, T)-characters as quantum cluster variables"],"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":"2026-07-22T22:24:34Z"}