Cluster Algebras of Type $D_4$, Tropical Planes, and the Positive Tropical Grassmannian

Abstract

We show that the number of combinatorial types of clusters of type $D_4$ modulo reflection-rotation is exactly equal to the number of combinatorial types of tropical planes in ${\mathbb{TP}}^5$. This follows from a result of Sturmfels and Speyer which classifies these tropical planes into seven combinatorial classes using a detailed study of the tropical Grassmannian $\mathrm{Gr}(3,6)$. Speyer and Williams show that the positive part ${\mathrm{Gr}}^{+}(3,6)$ of this tropical Grassmannian is combinatorially equivalent to a small coarsening of the cluster fan of type $D_4$. We provide a structural bijection between the rays of ${\mathrm{Gr}}^{+}(3,6)$ and the almost positive roots of type $D_4$ which makes this connection more precise. This bijection allows us to use the pseudotriangulations model of the cluster algebra of type $D_4$ to describe the equivalence of "positive" tropical planes in ${\mathbb{TP}}^5$, giving a combinatorial model which characterizes the combinatorial types of tropical planes using automorphisms of pseudotriangulations of the octogon.

16 pages, 12 figures, 2 tables