Hypersimplex

Abstract: 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.

Abstract: SVEN HERRMANN, MICHAEL JOSWIG, AND DAVID SPEYERAbstract. The Dressian Dr(k;n) parametrizes all tropical linear spaces, and it carriesa natural fan structure as a subfan of the secondary fan of the hypersimplex ( k;n). Weexplore the combinatorics of the rays of Dr(k;n), that is, the most degenerate tropicalplanes, for arbitrary k and n. This is related to a new rigidity concept for con gurationsof n k points in the tropical (k 1)-torus. Additional conditions are given for k = 3.On the way, we compute the entire fan Dr(3;8).

Abstract: There are $339$ combinatorial types of generic metrics on six points. They correspond to the $339$ regular triangulations of the second hypersimplex $\Delta(6,2)$, which also has $14$ non-regular triangulations.

Abstract: The Dressian Dr(k,n) parametrizes all tropical linear spaces, and it carries a natural fan structure as a subfan of the secondaryfan of the hypersimplex \Delta(k,n). We explore the combinatorics of the rays of Dr(k,n), that is, the most degenerate tropical planes, for arbitrary k and n. This is related to a new rigidity concept for configurations of n-k points in the tropical (k-1)-torus. Additional conditions are given for k=3. On the way, we compute the entire fan Dr(3,8).