On Cluster Algebras Arising from Unpunctured Surfaces

@article{Schiffler2007OnCA,
  title={On Cluster Algebras Arising from Unpunctured Surfaces},
  author={Ralf Schiffler and Hugh Thomas},
  journal={International Mathematics Research Notices},
  year={2007},
  volume={2009},
  pages={3160-3189}
}
We study cluster algebras that are associated to unpunctured surfaces, with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. In the special case where the cluster algebra is acyclic, we also give a formula for the… Expand

Figures and Tables from this paper

On cluster algebras arising from unpunctured surfaces II
We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of clusterExpand
Quantum cluster algebras from unpunctured triangulated surfaces
We study quantum cluster algebras from unpunctured surfaces, which are isomorphic to the skein algebras associated with the surfaces, [22], we generalize the Laurent expansion formula given by [31],Expand
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
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
Quantum cluster algebras from unpunctured triangulated surfaces: arbitrary coefficients and quantization
We study quantum cluster algebras from unpunctured surfaces with arbitrary coefficients and quantization. We first give a new proof of the Laurent expansion formulas for commutative cluster algebrasExpand
Unistructurality of cluster algebras of type A
Abstract It is conjectured by Assem, Schiffler and Shramchenko in [3] that every cluster algebra is unistructural, that is to say, that the set of cluster variables determines uniquely the clusterExpand
On cluster algebras from unpunctured surfaces with one marked point
We extend the construction of canonical bases for cluster algebras from unpunctured surfaces to the case where the number of marked points is one, and we show that the cluster algebra is equal to theExpand
Algebras from surfaces without punctures
We introduce a new class of finite dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts inExpand
Cluster algebras of unpunctured surfaces and snake graphs
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
Rooted Clusters for Graph LP Algebras
LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebrasExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 33 REFERENCES
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
A Cluster Expansion Formula (An case)
  • R. Schiffler
  • Mathematics, Computer Science
  • Electron. J. Comb.
  • 2008
TLDR
Given any seed $\Sigma$ in a Ptolemy cluster algebra, this work presents a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed $Sigma$. Expand
Denominators of cluster variables
Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables inExpand
Cluster algebras III: Upper bounds and double Bruhat cells
We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection ofExpand
From triangulated categories to cluster algebras II
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra \(\mathcal{A}\) of finite type can be realized asExpand
Cluster algebras and Weil-Petersson forms
In our previous paper we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case ofExpand
Quivers with Relations and Cluster Tilted Algebras
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. AnyExpand
A Graph Theoretic Expansion Formula for Cluster Algebras of Classical Type
In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type with bipartite seed. In particular, we provide aExpand
Cluster algebras II: Finite type classification
This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely manyExpand
Cluster algebras and triangulated surfaces. Part I: Cluster complexes
We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebraExpand
...
1
2
3
4
...