Resumen de Braid group symmetries of Grassmannian cluster algebras

Chris Fraser

• Let Gr ∘(k,n)⊂ Gr (k,n) denote the open positroid stratum in the Grassmannian. We define an action of the extended affine d-strand braid group on Gr ∘(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 Gr ∘(k,n), determining a homomorphism from the extended affine braid group to the cluster modular group for Gr (k,n). We also define a quasi-isomorphism between the Grassmannian Gr (k,rk) and the Fock–Goncharov configuration space of 2r-tuples of affine flags for SLk. 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 Gr (3,n) in terms of Kuperberg’s basis of non-elliptic webs. As our main application, we prove many of their conjectures for Gr (3,9) and give a presentation for its cluster modular group. We establish similar results for Gr (4,8). These results rely on the fact that both of these Grassmannians have finite mutation type.