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Title:Nakajima's (Q, T)-characters as quantum cluster variables
Author(s):Turmunkh, Bolor
Director of Research:Kedem, Rinat
Doctoral Committee Chair(s):Bergvelt, Maarten
Doctoral Committee Member(s):Di Francesco, Philippe; Nevins, Thomas
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Nakajima (q,t)-characters
Quantum cluster algebra
Abstract: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.
Issue Date:2017-05-03
Rights Information:Copyright 2017 Bolor Turmunkh
Date Available in IDEALS:2017-09-29
Date Deposited:2017-08

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