Bongartz Completion via c-Vectors
Bongartz Completion via c-Vectors
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DOI：
10.1093/imrn/rnac205
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发表时间：
2021-06
影响因子：
1
通讯作者：
P. Cao;Y. Gyoda;Toshiya Yurikusa
中科院分区：
文献类型：
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作者：
P. Cao;Y. Gyoda;Toshiya Yurikusa
本文首先利用$c$-向量给出了$\tau $-倾斜理论中Bongartz完备化的一个刻画。受此刻画的启发，我们用$c$-向量给出了簇代数中Bongartz完备化的定义。然后证明了簇代数中Bongartz完备化的存在唯一性。我们还证明了Bongartz完备化允许一定的可交换性。给出了Bongartz完备化在簇代数中的两个应用。作为第一个应用，我们证明了一个簇代数$\mathcal A$的交换图（或称为有向交换图）的完全子图同构于另一个簇代数的交换图，其中A $的顶点由包含特定簇变量的$\mathcal A$的种子组成.作为第二个应用，我们证明了在泛半域上的$Y$-模式中，每个$Y$-种子（直到一个$Y$-种子等价）是由这个$Y$-种子中的负$y$-变量唯一确定的。
In the present paper, we first give a characterization for Bongartz completion in $\tau $-tilting theory via $c$-vectors. Motivated by this characterization, we give the definition of Bongartz completion in cluster algebras using $c$-vectors. Then we prove the existence and uniqueness of Bongartz completion in cluster algebras. We also prove that Bongartz completion admits certain commutativity. We give two applications for Bongartz completion in cluster algebras. As the first application, we prove the full subquiver of the exchange quiver (or known as oriented exchange graph) of a cluster algebra $\mathcal A$ whose vertices consist of the seeds of $\mathcal A$ containing particular cluster variables is isomorphic to the exchange quiver of another cluster algebra. As the second application, we prove that in a $Y$-pattern over a universal semifield, each $Y$-seed (up to a $Y$-seed equivalence) is uniquely determined by the negative $y$-variables in this $Y$-seed.