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arXiv · 2609.23018

Unique mutation cycles of nucleus-extension quivers

Yuhang Li·Haiyan Zhu
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Abstract

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.

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Yuhang Li, Haiyan Zhu. 2026-09-19. Unique mutation cycles of nucleus-extension quivers. https://arxiv.org/abs/2609.23018

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