Nakajima's (Q, T)-characters as quantum cluster variables

Turmunkh, Bolor

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https://hdl.handle.net/2142/98093 Copy

Description

Title

Nakajima's (Q, T)-characters as quantum cluster variables

Author(s)

Turmunkh, Bolor

Issue Date

2017-05-03

Director of Research (if dissertation) or Advisor (if thesis)

Kedem, Rinat

Doctoral Committee Chair(s)

Bergvelt, Maarten

Committee Member(s)

• Di Francesco, Philippe
• Nevins, Thomas

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• T-system
• 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.

Graduation Semester

2017-08

Type of Resource

text

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http://hdl.handle.net/2142/98093

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Copyright 2017 Bolor Turmunkh

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