论文标题
阳性簇簇结构，来自重新标记的Plabic图
Positroid cluster structures from relabeled plabic graphs
论文作者
论文摘要
格拉斯曼尼亚人是开放式阳性品种$ p_v $的不一致联合，某些平稳的不可理用的亚变量，其定义是由完全阳性的动机。 $ p_v $的坐标环是群集代数，每个缩减的P_V $的Plabic Graph $ G $都决定了一个群集。我们研究将$ g $的边界顶点重新标记为排列$ r $的效果。在置换的合适假设下，我们证明了重新标记的图$ g^r $确定了其他开放式阳性式$ p_w $的群集。作为证据的关键步骤，我们表明$ P_V $和$ P_W $是非平凡的扭曲同构。我们的构造在每个开放式阳性变量$ p_w $上产生许多群集结构，并由具有适当重新标记的边界的Plabic图给出。我们猜想所有这些簇结构中的种子都是通过突变和涉及冷冻变量的劳伦斯单一变换的组合相关的，并为（开放的）Schubert和相反的Schubert品种建立了这种猜想。作为应用程序，我们还表明，对于某些降低的Plabic图$ G $，“源”群集和“目标”群集与突变和Laurent Monomial recalings相关。
The Grassmannian is a disjoint union of open positroid varieties $P_v$, certain smooth irreducible subvarieties whose definition is motivated by total positivity. The coordinate ring of $P_v$ is a cluster algebra, and each reduced plabic graph $G$ for $P_v$ determines a cluster. We study the effect of relabeling the boundary vertices of $G$ by a permutation $r$. Under suitable hypotheses on the permutation, we show that the relabeled graph $G^r$ determines a cluster for a different open positroid variety $P_w$. As a key step of the proof, we show that $P_v$ and $P_w$ are isomorphic by a nontrivial twist isomorphism. Our constructions yield many cluster structures on each open positroid variety $P_w$, given by plabic graphs with appropriately relabeled boundary. We conjecture that the seeds in all of these cluster structures are related by a combination of mutations and Laurent monomial transformations involving frozen variables, and establish this conjecture for (open) Schubert and opposite Schubert varieties. As an application, we also show that for certain reduced plabic graphs $G$, the "source" cluster and the "target" cluster are related by mutation and Laurent monomial rescalings.
