Crystal Structure of Upper Cluster Algebras

TL;DR

This paper introduces a crystal structure on upper cluster algebras, connecting algebraic, combinatorial, and categorical perspectives, and describes all biperfect bases parametrized by lattice points.

Contribution

It establishes a new crystal structure for upper cluster algebras and classifies all biperfect bases, providing a comprehensive algebraic and combinatorial framework.

Findings

Crystal structure can be algebraically lifted to the generic basis.

All biperfect bases are parametrized by lattice points in polyhedral sets.

Illustrated with classical and new examples.

Abstract

We describe the upper seminormal crystal structure for the $\mu$-supported $\delta$-vectors for any quiver with potential with reachable frozen vertices, or equivalently for the tropical points of the corresponding cluster $\mc{X}$-variety. We show that the crystal structure can be algebraically lifted to the generic basis of the upper cluster algebra. This can be viewed as an additive categorification of the crystal structure arising from cluster algebras. We introduce the biperfect bases in the cluster algebra setting and give a description of all biperfect bases, which are parametrized by lattice points in a product of polyhedral sets. We illustrate this theory from classical examples and new examples.