Setup: Let $B$ be a skew-symmetrisable integer matrix and $\mathcal{A}$ be the cluster algebra with principal coefficients at a seed with mutation matrix $\tilde{B}$ whose principal part is equal to $B$.
In [GHKK18], Gross, Hacking, Keel, Kontsevich associate to $\mathcal{A}$ a set of elements $\{\theta_q : q \in \mathcal{A}^\vee(\mathbb{Z}^T)\}$ called $\textit{theta functions}$. Here $\mathbb{Z}^T = (\mathbb{Z}, +, \max)$ and $\mathcal{A}^\vee$ is the $\mathcal{X}$-variety associated to $\tilde{B}^T$ (called the Fock-Goncharov dual in the Appendix A of [GHKK]).
Given any seed of $\mathcal{A}$, each theta function $\theta_q$ is a Laurent series in the extended cluster (i.e. in the cluster variables and frozen variables of that seed). Denote by $\Theta \subset \mathcal{A}^\vee(\mathbb{Z}^T)$ the set for which the corresponding theta functions are Laurent $\textit{polynomials}$ in some (equiv. any) seed of $\mathcal{A}$ (c.f. [GHKK, $\S 7$]).
Assumption: Assume that $\mathcal{A}$ is acyclic.
Questions:
- Is it true that $\Theta = \mathcal{A}^\vee(\mathbb{Z}^T)$ ?
- If the answer to 1) is yes, where can I find a reference?
Progress: $\textit{Under the assumption that B is skew-symmetric}$, [GHKK] tells us that
- Yes,
- Proposition 8.24 followed by Proposition 8.25 in [GHKK] (Corollary 8.30 followed by Proposition 8.28 in the arxiv version).
References: [GHKK] M. Gross, P. Hacking, S. Keel, M. Kontsevich, Canonical bases for cluster algebras. J. Amer. Math. Soc. 31 (2018), no. 2, 497-608, arxiv:1411.1394
