$\mathcal G$ -系统
$\mathcal G$-systems
摘要：
一个$\mathcal G$ -系统是$\mathbb Z^n$的$\mathbb Z$ -基的集合，带有一些额外的公理化条件.存在两种自然作用于$\mathcal G$ -系统的动作“突变”和“共Bongartz完成”，它们提供了$\mathcal G$ -系统的组合结构。结果表明，“co-Bongartz完备”与“突变”具有良好的相容性。“突变”的构造以前在不同的背景下是已知的，包括簇倾斜理论，淤积理论，$τ$ -倾斜理论，簇代数，标记曲面。我们发现，除了“突变”，还存在另一种行动“co-Bongartz完成”自然出现在这些不同的理论。借助于“co-Bongartz完备化”，可以很容易地得到一些好的组合结果。本文给出了不同理论下的“co-Bongartz完备化”的构造。然后我们证明了$\mathcal G$ -系统自然地产生于这些理论，并且不同理论中的“突变”和“co-Bongartz完备化”与$\mathcal G$ -系统中的“突变”和“co-Bongartz完备化”是相容的.
英文摘要：
A $\mathcal G$-system is a collection of $\mathbb Z$-bases of $\mathbb Z^n$ with some extra axiomatic conditions. There are two kinds of actions "mutations" and "co-Bongartz completions" naturally acting on a $\mathcal G$-system, which provide the combinatorial structure of a $\mathcal G$-system. It turns out that "co-Bongartz completions" have good compatibility with "mutations". The constructions of "mutations" are known before in different contexts, including cluster tilting theory, silting theory, $τ$-tilting theory, cluster algebras, marked surfaces. We found that in addition to "mutations", there exists another kind of actions "co-Bongartz completions" naturally appearing in these different theories. With the help of "co-Bongartz completions" some good combinatorial results can be easily obtained. In this paper, we give the constructions of "co-Bongartz completions" in different theories. Then we show that $\mathcal G$-systems naturally arise from these theories, and the "mutations" and "co-Bongartz completions" in different theories are compatible with those in $\mathcal G$-systems.
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