Density of g-Vector Cones From Triangulated Surfaces
Density of g-Vector Cones From Triangulated Surfaces
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DOI：
10.1093/imrn/rnaa008
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发表时间：
2019-04
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影响因子：
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通讯作者：
Toshiya Yurikusa
中科院分区：
文献类型：
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作者：
Toshiya Yurikusa
我们研究$g$-向量锥与簇代数的簇定义从一个标记表面$（S，M）$的秩$n$。我们确定封闭的联合$g$-向量锥与所有集群。它等于$\mathbb{R}^n$，除了一个闭曲面恰好有一个穿刺，在这种情况下，它等于$\mathbb{R}^n$中某个显式超平面的半空间。我们的主要成分是$（S，M）$，他们的剪切坐标和他们的渐近行为下德恩扭曲。作为一个应用，如果$（S，M）$不是一个具有恰好一个穿孔的闭曲面，则相应簇范畴中的簇倾斜对象的交换图是连通的。如果$（S，M）$是一个只有一个穿孔的闭曲面，它有两个连通分支。
We study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$ except for a closed surface with exactly one puncture, in which case it is equal to the half space of a certain explicit hyperplane in $\mathbb{R}^n$. Our main ingredients are laminations on $(S,M)$, their shear coordinates and their asymptotic behavior under Dehn twists. As an application, if $(S,M)$ is not a closed surface with exactly one puncture, the exchange graph of cluster tilting objects in the corresponding cluster category is connected. If $(S,M)$ is a closed surface with exactly one puncture, it has precisely two connected components.