Abstract. The (tree) amplituhedron A n,k,m is the image in the Grassmannian Gr k,k+m of the totally nonnegative part of Gr k,n , under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in N = 4 supersymmetric Yang-Mills theory. When k + m = n, the amplituhedron is isomorphic to the totally nonnegative Grassmannian, and when k = 1, the amplituhedron is a cyclic polytope. While the case m = 4 is most relevant to physics, the amplituhedron is an interesting mathematical object for any m. In this paper we study it in the case m = 1. We start by taking an orthogonal point of view and define a related "B-amplituhedron" B n,k,m , which we show is isomorphic to A n,k,m . We use this reformulation to describe the amplituhedron in terms of sign variation. We then give a cell decomposition of the amplituhedron A n,k,1 using the images of a collection of distinguished cells of the totally nonnegative Grassmannian. We also show that A n,k,1 can be identified with the complex of bounded faces of a cyclic hyperplane arrangement, and describe how its cells fit together. We deduce that A n,k,1 is homeomorphic to a ball.

We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is homeomorphic to a closed ball.

The (tree) amplituhedron \mathcal{A}_{n,k,m} is the image in the Grassmannian \mathrm {Gr}_{k,k+m} of the totally nonnegative Grassmannian \mathrm {Gr}_{k,n}^{\geq 0} , under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in planar \mathcal{N}=4 supersymmetric Yang–Mills theory. In the case relevant to physics ( m=4 ), there is a collection of recursively-defined 4k -dimensional BCFW cells in \mathrm {Gr}_{k,n}^{\geq 0} , whose images conjecturally "triangulate" the amplituhedron – that is, their images are disjoint and cover a dense subset of \mathcal{A}_{n,k,4} . In this paper, we approach this problem by first giving an explicit (as opposed to recursive) description of the BCFW cells. We then develop sign-variational tools which we use to prove that when k=2 , the images of these cells are disjoint in \mathcal{A}_{n,k,4} . We also conjecture that for arbitrary even m , there is a decomposition of the amplituhedron \mathcal{A}_{n,k,m} involving precisely M\big(k, n-k-m, \frac{m}{2}\big) top-dimensional cells (of dimension km ), where M(a,b,c) is the number of plane partitions contained in an a \times b \times c box. This agrees with the fact that when m=4 , the number of BCFW cells is the Narayana number N_{n-3,k+1} = \frac{1}{n-3}\binom{n-3}{k+1}\binom{n-3}{k} .

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We show that for each k and n, the cyclic shift map on the Grassmannian prefixGrk,nfalse(double-struckCfalse) has exactly ()nk fixed points. There is a unique totally nonnegative fixed point, given by taking n equally spaced points on the trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is even). We introduce a parameter q∈double-struckC×, and show that the fixed points of a q‐deformation of the cyclic shift map are precisely the critical points of the mirror‐symmetric superpotential Fq on prefixGrk,nfalse(double-struckCfalse). This follows from results of Rietsch about the quantum cohomology ring of prefixGrk,nfalse(double-struckCfalse). We survey many other diverse contexts which feature moment curves and the cyclic shift map.