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Register a new userCluster Algebras of Type $D_4$, Tropical Planes, and the Positive Tropical Grassmannian
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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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The well-known moment map maps the Grassmannian $Gr_{k+1,n}$ and the positive Grassmannian $Gr^+_{k+1,n}$ onto the hypersimplex $Delta_{k+1,n}$, which is a polytope of codimension $1$ inside $mathbb{R}^n$. Over the last decades there has been a great deal of work on matroid subdivisions (and positroid subdivisions) of the hypersimplex; these are closely connected to the tropical Grassmannian and positive tropical Grassmannian. Meanwhile any $n times (k+2)$ matrix $Z$ with maximal minors positive induces a map $tilde{Z}$ from $Gr^+_{k,n}$ to the Grassmannian $Gr_{k,k+2}$, whose image has full dimension $2k$ and is called the $m=2$ amplituhedron $A_{n,k,2}$. As the positive Grassmannian has a decomposition into positroid cells, one may ask when the images of a collection of cells of $Gr^+_{k+1,n}$ give a dissection of the hypersimplex $Delta_{k+1,n}$. By dissection, we mean that the images of these cells are disjoint and cover a dense subset of the hypersimplex, but we do not put any constraints on how their boundaries match up. Similarly, one may ask when the images of a collection of positroid cells of $Gr^+_{k,n}$ give a dissection of the amplituhedron $mathcal{A}_{n,k,2}$. In this paper we observe a remarkable connection between these two questions: in particular, one may obtain a dissection of the amplituhedron from a dissection of the hypersimplex (and vice-versa) by applying a simple operation to cells that we call the T-duality map. Moreover, if we think of points of the positive tropical Grassmannian $mbox{Trop}^+Gr_{k+1,n}$ as height functions on the hypersimplex, the corresponding positroidal subdivisions of the hypersimplex induce particularly nice dissections of the $m=2$ amplituhedron $mathcal{A}_{n,k,2}$. Along the way, we provide a new characterization of positroid polytopes and prove new results about positroidal subdivisions of the hypersimplex.
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