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 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.
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We are grateful to York University for hosting visits of the first and third authors. We also thank Hugh Thomas for helpful discussions.
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S. B. Brodsky was supported by the European Research Council grant SHPEF awarded to Olga Holtz. C. Ceballos was awarded partial support of the government of Canada through a Banting Postdoctoral Fellowship, of a York University research grant, and of the Austrian Science Foundation FWF, Grant F 5008-N15, in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”. J.-P. Labbé was supported by a FQRNT post-doctoral fellowship and a post-doctoral ISF Grant (805/11).
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Brodsky, S.B., Ceballos, C. & Labbé, JP. Cluster algebras of type \(D_4\), tropical planes, and the positive tropical Grassmannian. Beitr Algebra Geom 58, 25–46 (2017). https://doi.org/10.1007/s13366-016-0316-4
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DOI: https://doi.org/10.1007/s13366-016-0316-4