A Cluster Character with Coefficients for Cluster Category

Abstract

We introduce a cluster character with coefficients for a cluster category \(\mathcal {C}\) and rather than using a Frobenius 2-Calabi-Yau realization to incorporate coefficients into the representation-theoretic model for a cluster algebra, as done by Fu and Keller, we exploit intrinsic properties of \(\mathcal {C}\). For this purpose, we define an ice quiver associated to each cluster tilting object in \(\mathcal {C}\). In Dynkin case \(\mathbb {A}_{n}\), we also prove that the mutation class of the ice quiver associated to the cluster tilting object given by the direct sum of all projective objects is in bijection with set of ice quivers of cluster tilting objects in \(\mathcal {C}\) and that the study of a class of cluster algebra with coefficients can be reduced to the case that we called biframed.

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Correspondence to Tanise Carnieri Pierin.

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This work was inspired by the results obtained by the first author during his Ph.D. at Universidade de São Paulo, financially supported by Capes and CNPq, and under supervision of Prof. Eduardo do N. Marcos.

Presented by: Christof Geiss

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Borges, F., Pierin, T.C. A Cluster Character with Coefficients for Cluster Category. Algebr Represent Theor (2021). https://doi.org/10.1007/s10468-020-10016-8

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Keywords

  • Ice quiver
  • Cluster algebra
  • Cluster character

Mathematics Subject Classification (2010)

  • 13F60
  • 16G20