Linear recurrence relations for cluster variables of affine quivers

@article{Keller2010LinearRR,
  title={Linear recurrence relations for cluster variables of affine quivers},
  author={Bernhard Keller and Sarah Scherotzke},
  journal={arXiv: Representation Theory},
  year={2010}
}
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References

SHOWING 1-10 OF 38 REFERENCES
Cluster algebras as Hall algebras of quiver representations
Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be
Generalized associahedra via quiver representations
We provide a quiver-theoretic interpretation of certain smooth complete simplicial fans associated to arbitrary finite root systems in recent work of S. Fomin and A. Zelevinsky. The main properties
Cluster mutation-periodic quivers and associated Laurent sequences
We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the
Cluster algebras I: Foundations
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories
This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on
Q-system Cluster Algebras, Paths and Total Positivity ?
In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show
From triangulated categories to cluster algebras
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 as a
Friezes and a construction of the euclidean cluster variables
Generic Variables in Acyclic Cluster Algebras and Bases in Affine Cluster Algebras
Let $Q$ be a finite quiver without oriented cycles and $\mathcal A(Q)$ be the coefficient-free cluster algebra with initial seed $(Q,\textbf u)$. Using the Caldero-Chapoton map, we introduce and
...
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