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](https://arxiv.org/pdf/1501.04085.pdf) 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](https://arxiv.org/pdf/math/0602259.pdf) ) 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? 



Cluster algebras and tropical points
In short, yes. I would strongly recommend the survey by Nakanishi,


*Nakanishi, Tomoki*, [**Tropicalization method in cluster algebras**](http://dx.doi.org/10.1090/conm/580/11486), 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](https://zbmath.org/?q=an:1317.13053); [arXiv:1501.04085](https://arxiv.org/pdf/1501.04085.pdf)


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.


