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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 application/pdf Kedem, Rinat Bergvelt, Maarten Di Francesco, Philippe Nevins, Thomas Turmunkh, Bolor 2017-09-29T16:37:43Z 2017-09-29T16:37:43Z 2017-05-03 2017-08 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. http://hdl.handle.net/2142/98093 en Copyright 2017 Bolor Turmunkh T-system Nakajima (q,t)-characters Quantum cluster algebra Nakajima's (Q, T)-characters as quantum cluster variables text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.