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The enough g-pairs property and denominator vectors of cluster algebras

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Abstract

In this paper, we introduce the enough g-pairs property for principal coefficients cluster algebras, which can be understood as a strong version of the sign-coherence of the G-matrices. Then we prove that any skew-symmetrizable principal coefficients cluster algebra has the enough g-pairs property. As applications, we prove some long standing conjectures in cluster algebras, including a conjecture on denominator vectors and a conjecture on exchange graphs (see Conjectures 1, 2 below). In addition, we give a criterion to distinguish whether particular cluster variables belong to one common cluster for any skew-symmetrizable cluster algebra. As a corollary, we prove a conclusion which was conjectured by Fomin et al., cf. (Acta Math 201(1):83–146, 2008, Conjecture 5.5).

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Author notes

  1. Peigen Cao

    Present address: UFR de Mathématiques, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, Université de Paris, Batiment Sophie Germain, 75205, Paris Cedex 13, France

Authors and Affiliations

  1. Department of Mathematics, Zhejiang University (Yuquan Campus), Hangzhou, 310027, Zhejiang, People’s Republic of China

    Peigen Cao & Fang Li

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  1. Peigen Cao
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  2. Fang Li
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Correspondence to Fang Li.

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Communicated by Andreas Thom.

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This project is supported by the National Natural Science Foundation of China (No.11671350) and the Zhejiang Provincial Natural Science Foundation of China (Nos. LY19A010023 and LY18A010032).

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Cao, P., Li, F. The enough g-pairs property and denominator vectors of cluster algebras. Math. Ann. 377, 1547–1572 (2020). https://doi.org/10.1007/s00208-020-02033-1

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  • DOI: https://doi.org/10.1007/s00208-020-02033-1

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