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Cluster algebras arising from cluster tubes I: integer vectors

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Abstract

We study cluster algebras arising from cluster tubes. We obtain categorical interpretations for g-vectors, c-vectors and denominator vectors for cluster algebras of type \(\mathrm {C}\) with respect to arbitrary initial seeds. In particular, a denominator theorem has been proved, which enables us to establish the linearly independence of denominator vectors of cluster variables from the same cluster for cluster algebras of type \(\mathrm {A}\mathrm {B}\mathrm {C}\). This strengthens the link between cluster tubes and cluster algebras of type \(\mathrm {C}\) initiated by Buan, Marsh and Vatne.

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  1. In fact, Buan et al. [8] have considered the cluster algebra of type \(\mathrm {B}\) but not of type \(\mathrm {C}\). However, the cluster combinatorics of cluster algebras of type \(\mathrm {B}\) is the same as that of cluster algebras of type \(\mathrm {C}\) [23]. Hence here and after, we may state the result of [8] in cluster algebras of type \(\mathrm {C}\).

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Acknowledgements

We are very grateful to the anonymous referee for significant comments and corrections they proposed.

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Authors and Affiliations

  1. Department of Mathematics, SiChuan University, Chengdu, 610064, People’s Republic of China

    Changjian Fu & Shengfei Geng

  2. Department of Mathematics, Southwest Jiaotong University, Chengdu, 610031, People’s Republic of China

    Pin Liu

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  1. Changjian Fu
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  2. Shengfei Geng
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Correspondence to Shengfei Geng.

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To our teacher Liangang Peng on the occasion of his 60th birthday.

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Partially supported by the National Natural Science Foundation of China (no. 11471224).

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Fu, C., Geng, S. & Liu, P. Cluster algebras arising from cluster tubes I: integer vectors. Math. Z. 297, 1793–1824 (2021). https://doi.org/10.1007/s00209-020-02580-y

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