Cluster algebras and discrete integrable systems

This dissertation presents connections between cluster algebras and discrete integrable systems, especially T-systems and their specializations/generalizations. We give connections between the T-system or the octahedron relation, and the pentagram map and its various generalizations. A solution to the T-system with quasi-periodic boundary conditions gives rise to a solution to a higher pentagram map. In order to obtain all the solutions of higher pentagram map, we define T-systems with principal coefficients from cluster algebra aspect. Combinatorial solutions of the T-systems with principal coefficients with respect to any valid initial condition are shown to be partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2016). We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems of Goncharov and Kenyon 2013. We show that all Hamiltonians, partition functions of all weighted perfect matchings with a common homology class, are invariant under a move on the weighted graph. This move coincides with a cluster mutation, analog to Y-seed mutation in dimer integrable systems. Q-systems are reductions of T-systems by forgetting one of the parameters. We construct graphs for Q-systems of type A and B and show that the Hamiltonians are conserved quantities of the systems. The conserved quantities can be written as partition functions of hard particles on a certain graph. For type A, they Poisson commute under a nondegenerate Poisson bracket.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Vichitkunakorn, Panupong

Contributors dc:contributor

• Kedem, Rinat
• Di Francesco, Philippe
• Bergvelt, Maarten
• Yong, Alexander

• Cluster algebras
• Discrete integrable systems

Statement dc:rights

• Copyright 2017 Panupong Vichitkunakorn

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/97314

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/97314