Structural Properties of Reciprocal Generalized Cluster Algebras
2021-07
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Structural Properties of Reciprocal Generalized Cluster Algebras
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2021-07
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Ordinary cluster algebras were first introduced by Fomin and Zelevinsky in 2002 in order to provide a concrete combinatorial framework for studying dual canonical bases and total positivity in semisimple groups. Ordinary cluster algebras have since found applications in a wide array of areas, including: the representation theory of quivers, algebraic geometry and mirror symmetry, discrete integrable systems, Poisson geometry, Teichmuller theory, and mathematical physics. This unexpected ubiquity has made ordinary cluster algebras a natural object of interest for many mathematicians. In particular, there has been great interest in understanding their structural properties. One natural generalization of an ordinary cluster algebra is the generalized cluster algebra, introduced by Chekhov and Shapiro in 2013. In such algebras, the hallmark binomial exchange relations are replaced by polynomials of arbitrary degree. Given that there is a significant existing body of work about the structural properties of ordinary cluster algebras, it is natural to ask the same questions in the context of generalized cluster algebras. In particular, it is natural to ask if generalized cluster algebras exhibit positivity and if they have bases which are analogous to the various known bases for ordinary cluster algebras. In this thesis, we seek to understand these structural properties. We begin with the construction of generalized snake graphs, which extend the ordinary snake graphs of Musiker, Schiffler, and Williams to the setting of triangulated unpunctured orbifolds. We then use generalized snake graphs to establish cluster expansion formulas which associate cluster algebra elements to ordinary arcs, generalized arcs, and closed curves on triangulated orbifolds. As an immediate consequence, we obtain an alternate and explicitly combinatorial proof of positivity for such generalized cluster algebras. We also establish the notion of a universal snake graph, which can be used to recover both ordinary and generalized snake graphs. We then turn to cluster scattering diagrams and extend the cluster scattering diagram construction of Gross, Hacking, Keel, and Kontsevich to reciprocal generalized cluster algebras. We define generalized cluster varieties and verify that the definitions of useful objects such as broken lines and theta functions remain the same in the generalized setting. We then show that when the upper generalized cluster algebra and generalized cluster algebra coincide, this forms a basis for the generalized cluster algebra. Finally, we explicitly give the fixed data for the companion algebras associated to a particular generalized cluster algebra and explore properties of that data and the resulting cluster scattering diagrams.
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University of Minnesota Ph.D. dissertation. July 2021. Major: Mathematics. Advisor: Gregg Musiker. 1 computer file (PDF); vi, 145 pages.
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Kelley, Elizabeth. (2021). Structural Properties of Reciprocal Generalized Cluster Algebras. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/224991.
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