Positive tropical varieties and cluster algebras

Question: What is the connection between the tropicalization of the totally positive part of a variety, and the cluster algebra structure of the variety?

Background: In joint work with David Speyer, we have described the tropicalization of the totally positive part of the Grassmannian $G(k,n)$. When $k=2$, we get a fan which is closely related to the type $A$ associahedron. For $G(3,6)$ and $G(3,7)$, we get fans which are related to the type $D4$ and type $E6$ associahedra. Our results seem to be related to the results of Joshua Scott, who showed that the cluster algebra structure of the Grassmannians $G(2,n)$, $G(3,6)$, and $G(3,7)$ are of types $A$, $D4$, and $E6$, respectively.

(contributed by Lauren Williams)

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