核扩展箭图的唯一突变环
Unique mutation cycles of nucleus-extension quivers
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- School of Mathematical Sciences, Zhejiang University of Technology（浙江工业大学数学科学学院）
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中文总结 AI 辅助
本文证明了Fomin和Neville关于核扩展箭图长突变环的唯一性猜想，通过分析三种突变状态和子箭图，证明了每个核扩展箭图在带标签突变图中具有唯一简单环。
中文摘要 AI 辅助
我们证明了Fomin和Neville关于其长突变环在所有秩$n\ge4$中的唯一性猜想。更一般地，我们证明了每个核扩展箭图在其带标签的突变图中都有一个唯一的简单环。这些箭图允许在核外部有任意大小的无环部分。我们的证明使用了由闭核确定的三种突变状态，以及通过从环境箭图中一次删除核分解的一个部分而获得的完整子箭图。我们分析了这些状态之间的转换，以证明每个离开规定环的突变都在一个出口处执行。
英文摘要
We prove the uniqueness conjecture of Fomin and Neville for their long mutation cycles in all ranks $n\ge4$. More generally, we show that every nucleus-extension quiver has a unique simple cycle in its labeled mutation graph. These quivers allow an acyclic part of arbitrary size outside the nucleus. Our proof uses three mutation statuses determined by a closed nucleus and the full subquivers obtained by deleting one part of its nucleus decomposition at a time from the ambient quiver. We analyze transitions between these statuses to prove that every mutation leaving the prescribed cycle is performed at an exit.
