Results 1 
4 of
4
Universal geometric cluster algebras
, 2012
"... We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually Z, Q, or R. We broaden the definition of geometric cluster algebras slightly over the us ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually Z, Q, or R. We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over B with universal geometric coefficients, or the universal geometric cluster algebra over B. Constructing universal geometric coefficients is equivalent to finding an Rbasis for B (a “mutationlinear” analog of the usual linearalgebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan FB, which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between FB and gvectors. We construct universal geometric coefficients in rank 2 and in finite type and discuss the construction in affine type.
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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
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
Contents
"... Abstract. Cluster variables in a cluster algebra can be parametrized by two families of integer vectors: dvectors and gvectors. While gvectors satisfy two recursive formulas (one for initialseedmutations and one for finalseedmutations), dvectors admit only a finalseedmutation recursion. We ..."
Abstract
 Add to MetaCart
Abstract. Cluster variables in a cluster algebra can be parametrized by two families of integer vectors: dvectors and gvectors. While gvectors satisfy two recursive formulas (one for initialseedmutations and one for finalseedmutations), dvectors admit only a finalseedmutation recursion. We present an initialseedmutation formula for dvectors and give two rephrasings of this recursion: one as a duality formula for dvectors in the style of the gvectors/cvectors dualities of Nakanishi and Zelevinsky, and one as a formula expressing the highest powers in the Laurent expansion of a cluster variable in terms of the dvectors of any cluster containing it. We prove that the initialseedmutation recursion holds in a varied collection of cluster algebras, but not in general. We conjecture further that the formula holds for sourcesink moves on the initial seed in an arbitrary cluster algebra, and we prove this conjecture in the case of surfaces.