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Cluster algebras of type \(D_4\), tropical planes, and the positive tropical Grassmannian. (English) Zbl 1401.13063
Summary: 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.

13F60 Cluster algebras
14T05 Tropical geometry (MSC2010)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
52C30 Planar arrangements of lines and pseudolines (aspects of discrete geometry)
polymake; SageMath
Full Text: DOI
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