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arXiv · 1511.02699

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

Sarah B. Brodsky·Cesar Ceballos·Jean-Philippe Labbé
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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 $\operatorname{Gr}(3,6)$. Speyer and Williams show that the positive part $\operatorname{Gr}^+(3,6)$ 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 $\operatorname{Gr}^+(3,6)$ 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.

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Sarah B. Brodsky, Cesar Ceballos, Jean-Philippe Labbé. 2015-11-09. Cluster Algebras of Type $D_4$, Tropical Planes, and the Positive Tropical Grassmannian. https://arxiv.org/abs/1511.02699

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