Crystal Structure of Upper Cluster Algebras

We describe the (weaker) upper seminormal crystal structure for the \(\mu\)-supported \(\delta\)-vectors for any ice quiver with potential, or equivalently for the tropical points of the corresponding cluster \(\mathcal{X}\)-variety. We show that the crystal structure can be algebraically lifted to a biperfect basis of the upper cluster algebra. This can be viewed as an additive categorification of the crystal structure arising from cluster algebras. All such biperfect bases are parametrized by lattice points in a product of polytopes. We find that the requirement for upgrading to a (semi)normal crystal is almost minimal in some sense. We illustrate this theory from classical examples and new examples.