Cluster algebras and tropical points

Question

Recently, I have been looking at some articles about bases for cluster algebras and came across the idea of tropical points. I should highlight here that unfortunately I have no background on algebraic geometry (and in what follows, tropical geometry) and so I have been first trying to understand what is the equivalent of tropical points on the cluster algebra side of things. In here the author introduces tropical transformation (Def. 3.2.1) which is the piecewise linear map which takes tropical points $D(t)$ corresponding to a seed $t$, to tropical points $D(\mu_k t)$ corresponding to a seed $\mu_k t$, where $\mu_k$ is the mutation of one of the elements from the seed in direction $k$. Now for the cluster algebra point of view, the author remarks that Fomin and Zelevinsky wrote this transformation ( (7.18) in here ) and conjectured that it describes the transformation of the $g$-vectors of cluster variables. It would suggests that I can think of tropical points as $g$-vectors of cluster variables? Is that at all correct?

Answer 1

In short, yes. I would strongly recommend the survey by Nakanishi,

Nakanishi, Tomoki, Tropicalization method in cluster algebras, Athorne, Chris (ed.) et al., Tropical geometry and integrable systems. A conference on tropical geometry and integrable systems, School of Mathematics and Statistics, Glasgow, UK, July 3–8, 2011. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-7553-7/pbk; 978-0-8218-9188-9/ebook). Contemporary Mathematics 580, 95-115 (2012). ZBL1317.13053; arXiv:1501.04085

which explains how tropical geometry relates to cluster algebras, although should you need to go deeper you will need to explore the references therein (especially the influential papers of Fock--Goncharov) and possibly more recent work, e.g. that of Gross--Hacking--Keel--Kontsevich.