  We define an action of the extended affine d-strand braid group on the open
positroid stratum in the Grassmannian Gr(k,n), for d the greatest common
divisor of k and n. The action is by quasi-automorphisms of the cluster
structure on the Grassmannian, determining a homomorphism from the extended
affine braid group to the cluster modular group. 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 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
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.
