For any quiver mutation sequence, we define a pair of matrices that describe a fixed point equation of a cluster transformation determined from the mutation sequence. We give an explicit relationship between this pair of matrices and the Jacobian matrix of the cluster transformation. Furthermore, we show that this relationship reduces to a relationship between the pair of matrices and the $C$-matrix of the cluster transformation in a certain limit of cluster variables. As an application, we prove that quivers associated with once-punctured surfaces do not have maximal green or reddening sequences.

中文翻译：

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Y 种子突变的雅可比矩阵

对于任何 quiver 突变序列，我们定义一对矩阵来描述从突变序列确定的聚类变换的不动点方程。我们给出了这对矩阵和聚类变换的雅可比矩阵之间的明确关系。此外，我们表明这种关系简化为矩阵对与集群变换的 $C$-矩阵之间的关系，在一定的集群变量限制下。作为一个应用，我们证明与一次刺破表面相关的箭袋没有最大的绿色或变红序列。