Generic bases for cluster algebras from the cluster category

By Pierre-Guy Plamondon. 2011

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Inspired by recent work of Geiss-Leclerc-Schroer, we use Hom-finite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by the cluster character on objects having the same index. If the matrix associated to the quiver is of full rank, then we prove that the elements in this set are linearly independent. If the cluster algebra arises from the setting of Geiss-Leclerc-Schroer, then we obtain the basis found by these authors. We show how our point of view agrees with the spirit of conjectures of Fock-Goncharov concerning the parametrization of a basis of the upper cluster algebra by points in the tropical X-variety.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d2e89f990482ec9dc6cb98f91b8b09d87a0f2df1

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Citations in the corpus

Greedy elements in rank 2 cluster algebras (2012, background): “. A lot of recent activity in the ﬁeld has been directed towards various constructions of “natural” bases in cluster algebras. An overview of these approaches with relevant references can be found in [9]. This paper builds upon the approach developed in [11], where it was shown that the indecomposab”