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Unique mutation cycles of nucleus-extension quivers
Abstract
We prove the uniqueness conjecture of Fomin and Neville for their long mutation cycles in all ranks n≥4. 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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BibTeX
@misc{li2026unique,
title = {{Unique mutation cycles of nucleus-extension quivers}},
author = {Yuhang Li and Haiyan Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23018},
eprint = {2609.23018},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23018},
}
RIS
TY - UNPB
TI - Unique mutation cycles of nucleus-extension quivers
AU - Yuhang Li
AU - Haiyan Zhu
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23018
UR - https://arxiv.org/abs/2609.23018
AB - We prove the uniqueness conjecture of Fomin and Neville for their long mutation cycles in all ranks n≥4. 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.
ER -