Cluster automorphism group of braid varieties

The cluster automorphism group of a cluster variety was defined by Gekhtman–Shapiro–Vainshtein, and later studied by Lam–Speyer. Braid varieties are interesting affine algebraic varieties indexed by positive braid words. It was proved recently that braid varieties are cluster varieties. In this paper, we propose a description of the cluster automorphism group and its action on braid varieties, and compute several examples.