Results 1 -
4 of
4
Universal geometric coefficients for the once-punctured torus
, 2012
"... Abstract. We construct universal geometric coefficients, over Z, Q, and R, for cluster algebras arising from the once-punctured torus. We verify that the once-punctured torus has a property called the Null Tangle Property. The universal geometric coefficients over Z and Q are then given by the shear ..."
Abstract
-
Cited by 4 (3 self)
- Add to MetaCart
(Show Context)
Abstract. We construct universal geometric coefficients, over Z, Q, and R, for cluster algebras arising from the once-punctured torus. We verify that the once-punctured torus has a property called the Null Tangle Property. The universal geometric coefficients over Z and Q are then given by the shear co-ordinates of certain “allowable ” curves in the torus. The universal geometric coefficients over R are given by the shear coordinates of allowable curves to-gether 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 once-punctured torus and recover a result of Nájera on g-vectors. 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 re-lated by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cl ..."
Abstract
-
Cited by 3 (2 self)
- Add to MetaCart
(Show Context)
Abstract. A universal geometric cluster algebra over an exchange matrix B is a universal object in the category of geometric cluster algebras over B re-lated by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster al-gebras 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 once-punctured 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 g-vector fan associated to any exchange matrix B whose Cartan companion is of finite or affine type, using the combinatorics and geometry of Coxeter-sortable elements and Cambrian lat ..."
Abstract
- Add to MetaCart
This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and g-vector fan associated to any exchange matrix B whose Cartan companion is of finite or affine type, using the combinatorics and geometry of Coxeter-sortable elements and Cambrian lattices/fans. Specifically, we construct a framework in the unique non-acyclic affine case, the cyclically ori-ented n-cycle. 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 n-cycle, using a more general notion of sortable elements for quivers with cycles.
