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Showing 1 to 10 of 10 for “"Cluster algebras"”.
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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 …
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Cluster algebras from surfaces and their bases
Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-20 without embargo terms
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Cluster Algebras and Maximal Green Sequences for Closed Surfaces
… For each triangulation, T, we can associate a cluster algebra. In this paper we will consider orientable surfaces of genus n with two interior marked points and no boundary component. We will construct a specific triangulation of this surface which yields a quiver. Then in the sense of work by …
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MUTATION INVARIANT FUNCTIONS ON CLUSTER ENSEMBLES ASSOCIATED WITH SURFACES
… define the notion of an invariant function on a cluster ensemble with respect to a group action of the cluster modular group on its associated function fields. We realize many examples of previously studied functions as elements of this type of invariant ring and give many new examples. We …
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Applications of graded methods to cluster variables in arbitrary types
… properties of gradings on several examples of cluster algebras, primarily of infinite type. We start by considering two classes of finite type cluster algebras: those of type Bn and Cn. We give the number of cluster variables of each occurring degree and verify that the grading is balanced. …
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Properties of Polynomial Identity Quantized Weyl Algebras
… work on Polynomial Identity (PI) quantized Weyl algebras we begin with a brief survey of Poisson geometry and quantum cluster algebras, before using these as tools to classify the possible centers of such algebras in two different ways. In doing so we explicitly calculate the formulas of the …
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Combinatorial aspects of total positivity
… between positive tropical varieties and the cluster algebras of Fomin and Zelevinsky, proving the conjecture in the case of the Grassmannian. The third and fourth parts of my thesis explore a notion of total positivity for oriented matroids. Namely, I introduce the positive Bergman complex of …
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Categorifications of Cluster Varieties
… varieties whose coordinate rings admit a cluster algebra structure. The presence of a cluster structure has important implications for the geometry of a variety including the existence of canonical linearly independent sets of regular functions. Additionally, methods for studying cluster …
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Cluster Structure for Mirkovic-Vilonen Cycles and Polytopes
… Mirkovic-Vilonen (MV) basis for semisimple Lie algebras and compare this to the associated cluster algebra to investigate the question of whether or not the cluster variables are in the MV basis. We begin with finding analogues of the cluster structure among MV cycles and MV polytopes. In …
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Generalized Bäcklund-Darboux transformations for Coxeter-Toda systems on simple Lie groups
… Coxeter-Toda systems on simple Lie groups from a cluster algebraic prospective, using the cluster structure on Coxeter double Bruhat cells developed by Fock-Goncharov, Gekhtman et al. and Williams, thereby generalizing the construction developed by Gekhtman et al. We will then show that these …