Braid group symmetries of Grassmannian cluster algebras

Abstract

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.