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Positivity for cluster algebras from surfaces
We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitraryExpand
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Bases for cluster algebras from surfaces
We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, such asExpand
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Cluster expansion formulas and perfect matchings
We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in theseExpand
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Linear systems on tropical curves
A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear systemExpand
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Matrix formulae and skein relations for cluster algebras from surfaces
This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given anyExpand
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Higher cluster categories and QFT dualities
We present a unified mathematical framework that elegantly describes minimally SUSY gauge theories in even dimension, ranging from $6d$ to $0d$, and their dualities. This approach combines recentExpand
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Aztec Castles and the dP3 Quiver
Bipartite, periodic, planar graphs known as brane tilings can be associated to a large class of quivers. This paper will explore new algebraic properties of the well-studied del Pezzo 3 quiver andExpand
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A COMPENDIUM ON THE CLUSTER ALGEBRA AND QUIVER PACKAGE IN Sage
This is the compendium of the cluster algebra and quiver package for Sage. The purpose of this package is to provide a platform to work with cluster algebras in graduate courses and to furtherExpand
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Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type
TLDR
Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the $(b,c)$ family, possesses the Laurentness property: for all positive integers b and c, the sequence of Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs. Expand
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Gale-Robinson Sequences and Brane Tilings
We study variants of Gale-Robinson sequences, as motivated by cluster algebras with principal coefficients. For such cases, we give combinatorial interpretations of cluster variables using braneExpand
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