Appears in collection : Combinatorics and Arithmetic for Physics: special days 2023

Crystal operators on canonical bases as introduced by Kashiwara/Lusztig provide in particular a toolbox to compute within the category of finite dimensional representations of finite dimensional simple Lie algebras. Motivated by this we previously introduced certain operators on the lattice of tropical points of mirror dual A- and X-cluster spaces. In this talk we give an update. In particular, the crystal structure gives rise to a binary operation on the canonical basis due to Gross-Hacking-Keel-Kontsevich. We expect this to have a wider range of applications in the theory of cluster algebras and in physics.