Let \({{\,\mathrm{\text{ Gr }}\,}}^\circ (k,n) \subset {{\,\mathrm{\text{ Gr }}\,}}(k,n)\) denote the open positroid stratum in the Grassmannian. We define an action of the extended affine d-strand braid group on \({{\,\mathrm{\text{ Gr }}\,}}^\circ (k,n)\) by regular automorphisms, for d the greatest common divisor of k and n. The action is by quasi-automorphisms of the cluster structure on \({{\,\mathrm{\text{ Gr }}\,}}^\circ (k,n)\), determining a homomorphism from the extended affine braid group to the cluster modular group for \({{\,\mathrm{\text{ Gr }}\,}}(k,n)\). We also define a quasi-isomorphism between the Grassmannian \({{\,\mathrm{\text{ Gr }}\,}}(k,rk)\) and the Fock–Goncharov configuration space of 2r-tuples of affine flags for \({{\,\mathrm{\text {SL}}\,}}_k\). This identifies the cluster variables, clusters, and cluster modular groups, in these two cluster structures. Fomin and Pylyavskyy proposed a description of the cluster combinatorics for \({{\,\mathrm{\text{ Gr }}\,}}(3,n)\) in terms of Kuperberg’s basis of non-elliptic webs. As our main application, we prove many of their conjectures for \({{\,\mathrm{\text{ Gr }}\,}}(3,9)\) and give a presentation for its cluster modular group. We establish similar results for \({{\,\mathrm{\text{ Gr }}\,}}(4,8)\). These results rely on the fact that both of these Grassmannians have finite mutation type.

中文翻译：

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格拉斯曼簇代数的辫子群对称

令\（{{\，\ mathrm {\ text {Gr}} \，}} ^ \ circ（k，n）\ subset {{\，\ mathrm {\ text {Gr}} \，}}（k， n）\）表示格拉斯曼方程中的开放正片状层。我们定义扩展的仿射的动作d -链辫群上\（{{\，\ mathrm {\文本{的Gr}} \，}} ^ \ CIRC（K，N）\）通过定期构，为d的k和n的最大公约数。该动作是通过\（{{\，\ mathrm {\ text {Gr}} \，}} ^ \ circ（k，n）\）上的簇结构的拟自同构来确定的，并根据扩展的仿射编织确定同构组到\（{{\，\ mathrm {\ text {Gr}} \，}}（k，n）\）的集群模块化组。我们还定义了Grassmannian之间的准同构\（{{\，\ mathrm {\ text {Gr}} \，}}（k，rk）\）和Fock–Goncharov配置空间为2 r- \（{{ \\\ mathrm {\ text {SL}} \，}} _ k \）。这将标识这两个集群结构中的集群变量，集群和集群模块化组。Fomin和Pylyavskyy提出了基于Kuperberg非椭圆网的\（{{\\ mathrm {\ text {Gr}} \，}}（3，n）\）的聚类组合描述。作为我们的主要应用，我们证明了他们对\（{{\，\ mathrm {\ text {Gr}} \，}}（3,9）\）的许多猜想，并对其群集模块组进行了介绍。我们为\（{{\，\ mathrm {\ text {Gr}} \，}}（4,8）\）建立相似的结果。这些结果依赖于这两个格拉斯曼主义者都具有有限突变类型的事实。