AI 中文总结
本文证明了Fraser--Sherman-Bennett猜想，通过构造倾斜模和Frobenius范畴化，建立了重标记正丛簇结构的拟等价性，并利用导出等价和幺模变换证明了坐标环上的恒等映射是拟簇同构。
AI 中文摘要
我们证明了Fraser--Sherman-Bennett关于由允许重标记的plabic图产生的正丛簇结构拟等价性的猜想。我们从边界项链构造了一个倾斜模，并获得了重标记簇结构的Frobenius范畴化。通过将秩一模的局部化特征与Plücker坐标等同，我们将可变面模的稳定像与稳定范畴中可达簇倾斜对象的和项相匹配。由此导出的导出等价产生了相对指标格的幺模变换，该变换交织了完整的扩展交换矩阵。这些比较表明，坐标环上的恒等映射是两个簇结构之间的拟簇同构。
英文摘要
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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