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Relation Between f-Vectors and d-Vectors in Cluster Algebras of Finite Type or Rank 2

Abstract

We study f-vectors, which are the maximal degree vectors of F-polynomials in cluster algebra theory. For a cluster algebra of finite type, we find that positive f-vectors correspond with d-vectors, which are exponent vectors of denominators of cluster variables. Furthermore, using this correspondence and properties of d-vectors, we prove that cluster variables in a cluster are uniquely determined by their f-vectors when the cluster algebra is of finite type or rank 2.

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Acknowledgements

The author would like to express his gratitude to Bernhard Keller for insightful comments about Theorem 1.8. The author appreciates important remarks about Conjecture 1.10 by Changjian Fu. Toshiya Yurikusa gives helpful advice about Theorem 1.11. The author received generous support from Tomoki Nakanishi. The author also thanks Haruhisa Enomoto, Yoshiki Aibara, and Naohiro Tsuzu. This work was supported by JSPS KAKENHI Grant number JP20J12675.

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Correspondence to Yasuaki Gyoda.

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Gyoda, Y. Relation Between f-Vectors and d-Vectors in Cluster Algebras of Finite Type or Rank 2. Ann. Comb. 25, 573–594 (2021). https://doi.org/10.1007/s00026-021-00527-6

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Mathematics Subject Classification

  • 13F60

Keywords

  • Cluster algebra
  • Mutation
  • F-vector
  • D-vector