The enoughg-pairs property and denominator vectors of cluster algebras
The enoughg-pairs property and denominator vectors of cluster algebras
复制标题
簇代数的oughg-pairs属性和分母向量
DOI：
10.1007/s00208-020-02033-1
复制
发表时间：
2020
影响因子：
1.4
通讯作者：
Li Fang
中科院分区：
文献类型：
--
作者：
Cao Peigen;Li Fang
在本文中，我们介绍了主系数簇代数的足够g对性质，它可以被理解为G矩阵符号相干性的强版本。然后我们证明任何可偏斜对称的主系数簇代数都具有足够的g对性质。作为应用，我们证明了簇代数中一些长期存在的猜想，包括分母向量的猜想和交换图的猜想（参见下面的猜想 1、2）。此外，我们给出了一个标准来区分特定簇变量是否属于任何可偏斜对称簇代数的一个公共簇。作为推论，我们证明了 Fomin 等人推测的结论，参见。 （《数学学报》201(1)：83–146，2008 年，猜想 5.5）。
In this paper, we introduce the enoughg-pairs property for principal coefficients cluster algebras, which can be understood as a strong version of the sign-coherence of theG-matrices. Then we prove that any skew-symmetrizable principal coefficients cluster algebra has the enoughg-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).