Braid group symmetries of Grassmannian cluster algebras

Chris Fraser

Published 2017-02-01Version 1

We define an action of the k-strand braid group on the set of cluster variables for the Grassmannian Gr(k,n), whenever k divides n. The action sends clusters to clusters, preserving the underlying quivers, defining a homomorphism from the braid group to the cluster modular group for Gr(k,n). The action is induced by certain rational maps on Gr(k,n). Each of these maps is a quasi-automorphism of the cluster structure, a notion we defined in previous work. We also describe a quasi-isomorphism between Gr(k,n) and a certain Fock-Goncharov space of SL_k-local systems in a disk. The quasi-isomorphism identifies the cluster variables, clusters, and cluster modular groups, in these two cluster structures. Fomin and Pylyavskyy have 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 these conjectures for Gr(3,9). The proof relies on the fact that Gr(3,9) is of finite mutation type.