文摘
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 generic tropical planes in \(\mathbb {TP}^5\). This follows from a result of Sturmfels and Speyer which classifies these generic 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” generic tropical planes in \(\mathbb {TP}^5\), giving a combinatorial model which characterizes the combinatorial types of generic tropical planes using automorphisms of pseudotriangulations of the octogon.