Abstract

The sign coherence phenomenon is an important feature of c-vectors in cluster algebras with principal coefficients. In this note, we consider a more general version of c-vectors defined for arbitrary cluster algebras of geometric type and formulate a conjecture describing their asymptotic behavior. This conjecture, which is called the asymptotic sign coherence conjecture, states that for any infinite sequence of matrix mutations that satisfies certain natural conditions, the corresponding c-vectors eventually become sign coherent. We prove this conjecture for rank 2 cluster algebras of infinite type and for a particular sequence of mutations in a cluster algebra associated with the Markov quiver.

摘要

符号相干现象是主系数簇代数中c向量的一个重要特征。在本笔记中，我们考虑为任意几何类型的簇代数定义的更一般的c向量版本，并制定描述其渐近行为的猜想。这个猜想被称为渐近符号相干猜想，它指出对于任何满足某些自然条件的矩阵突变的无限序列，相应的c向量最终变得符号相干。我们为无限类型的秩 2 簇代数和与马尔可夫箭袋相关的簇代数中的特定突变序列证明了这一猜想。