In the paper [1], the authors define functions on double Bruhat cells, which they call generalized minors. By relating certain double Bruhat cells to the spaces $\operatorname {Conf}_3 {\mathcal {A}}_G$ and $\operatorname {Conf}_4 {\mathcal {A}}_G$, we give formulas for these generalized minors as tensor invariants. This allows us to verify certain weight identities conjectured in [7]. We then show that the grading of these tensor invariants is equivalent information to the quiver for the cluster structure on $\operatorname {Conf}_3 {\mathcal {A}}_G$. This leads to a simple combinatorial construction of cluster structures on the moduli space of framed local systems ${\mathcal {X}}_{G^{\prime},S}$ and ${\mathcal {A}}_{G,S}$.

中文翻译：

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广义次要和张量不变量

在论文 [1] 中，作者在双布鲁哈特细胞上定义了函数，他们称之为广义未成年人。通过将某些双布鲁哈特细胞与空间 $\operatorname {Conf}_3 {\mathcal {A}}_G$ 和 $\operatorname {Conf}_4 {\mathcal {A}}_G$ 相关联，我们给出了这些广义未成年人的公式作为张量不变量。这使我们能够验证 [7] 中推测的某些权重恒等式。然后，我们证明这些张量不变量的分级与 $\operatorname {Conf}_3 {\mathcal {A}}_G$ 上的集群结构的箭袋是等价的信息。这导致在框架局部系统 ${\mathcal {X}}_{G^{\prime},S}$ 和 ${\mathcal {A}}_{G ,S}$。