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PreprintCC BY 4.0arXiv

Quasi-equivalence of relabeled positroid cluster structures

  • Yilin Wu
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Abstract

We prove the Fraser--Sherman-Bennett conjecture on the quasi-equivalence of positroid cluster structures arising from admissibly relabeled plabic graphs. We construct a tilting module from the boundary necklace and obtain a Frobenius categorification of the relabeled cluster structure. By identifying localized characters of rank-one modules with Plücker coordinates, we match the stable images of the mutable face modules with the summands of a reachable cluster-tilting object in the stable category. The induced derived equivalence yields a unimodular transformation of relative index lattices intertwining the full extended exchange matrices. These comparisons show that the identity map on the coordinate ring is a quasi-cluster isomorphism between the two cluster structures.

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Cite

BibTeXRIS
BibTeX
@misc{wu2026quasiequivalence,
  title = {{Quasi-equivalence of relabeled positroid cluster structures}},
  author = {Yilin Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.30637},
  eprint = {2609.30637},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30637},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Quasi-equivalence of relabeled positroid cluster structures
AU  - Yilin Wu
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.30637
UR  - https://arxiv.org/abs/2609.30637
AB  - We prove the Fraser--Sherman-Bennett conjecture on the quasi-equivalence of positroid cluster structures arising from admissibly relabeled plabic graphs. We construct a tilting module from the boundary necklace and obtain a Frobenius categorification of the relabeled cluster structure. By identifying localized characters of rank-one modules with Plücker coordinates, we match the stable images of the mutable face modules with the summands of a reachable cluster-tilting object in the stable category. The induced derived equivalence yields a unimodular transformation of relative index lattices intertwining the full extended exchange matrices. These comparisons show that the identity map on the coordinate ring is a quasi-cluster isomorphism between the two cluster structures.
ER  -