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Universal geometric coefficients for the oncepunctured torus
, 2012
"... Abstract. We construct universal geometric coefficients, over Z, Q, and R, for cluster algebras arising from the oncepunctured torus. We verify that the oncepunctured torus has a property called the Null Tangle Property. The universal geometric coefficients over Z and Q are then given by the shear ..."
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Abstract. We construct universal geometric coefficients, over Z, Q, and R, for cluster algebras arising from the oncepunctured torus. We verify that the oncepunctured torus has a property called the Null Tangle Property. The universal geometric coefficients over Z and Q are then given by the shear coordinates of certain “allowable ” curves in the torus. The universal geometric coefficients over R are given by the shear coordinates of allowable curves together with the normalized shear coordinates of certain other curves each of which is dense in the torus. We also construct the mutation fan for the oncepunctured torus and recover a result of Nájera on gvectors. Contents
Universal geometric cluster algebras from surfaces
 Trans. Amer. Math. Soc
"... Abstract. A universal geometric cluster algebra over an exchange matrix B is a universal object in the category of geometric cluster algebras over B related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cl ..."
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Abstract. A universal geometric cluster algebra over an exchange matrix B is a universal object in the category of geometric cluster algebras over B related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster algebras relative to the definition originally given Fomin and Zelevinsky.) The universal objects are closely related to a fan FB called the mutation fan for B. In this paper, we consider universal geometric cluster algebras and mutation fans for cluster algebras arising from marked surfaces. We identify two crucial properties of marked surfaces: The Curve Separation Property and the Null Tangle Property. The latter property implies the former. We prove the Curve Separation Property for all marked surfaces except oncepunctured surfaces without boundary components, and as a result we obtain a construction of the rational part of FB for these surfaces. We prove the Null Tangle Property for a smaller family of surfaces, and use it to construct universal geometric coefficients for these surfaces. Contents
A Cambrian framework for the oriented cycle
, 2015
"... This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and gvector fan associated to any exchange matrix B whose Cartan companion is of finite or affine type, using the combinatorics and geometry of Coxetersortable elements and Cambrian lat ..."
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This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and gvector fan associated to any exchange matrix B whose Cartan companion is of finite or affine type, using the combinatorics and geometry of Coxetersortable elements and Cambrian lattices/fans. Specifically, we construct a framework in the unique nonacyclic affine case, the cyclically oriented ncycle. In the acyclic affine case, a framework was constructed by combining a copy of the Cambrian fan for B with an antipodal copy of the Cambrian fan for −B. In this paper, we extend this “doubled Cambrian fan” construction to the oriented ncycle, using a more general notion of sortable elements for quivers with cycles.