# Braid group symmetries of Grassmannian cluster algebras

## TL;DR

This paper explores the symmetries of Grassmannian cluster algebras via braid group actions, establishing connections with configuration spaces and proving conjectures for specific cases.

## Contribution

It introduces a braid group action on Grassmannian cluster structures, links Grassmannians with configuration spaces, and proves conjectures for Gr(3,9) and Gr(4,8).

## Key findings

- Defined braid group actions on Grassmannian cluster structures.
- Established isomorphisms between Grassmannians and configuration spaces.
- Proved conjectures for Gr(3,9) and Gr(4,8).

## Abstract

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.