# Bases for upper cluster algebras and tropical points

## TL;DR

This paper characterizes all bases containing cluster monomials and parametrized by tropical points for a broad class of upper cluster algebras, establishing the existence of a generic basis and linking theta functions to Bridgeland's formula.

## Contribution

It provides a complete description of bases with specific properties for injective-reachable upper cluster algebras with full rank coefficients, including the existence of a generic basis and effectiveness of Bridgeland's formula.

## Key findings

- All such bases are described for the class of algebras considered.
- Existence of a generic basis is established.
- Bridgeland's formula effectively computes theta functions.

## Abstract

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of its bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland's representation theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all well-known cluster algebras arising from representation theory or higher Teichm\"uller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds.