#####
Séminaire Lotharingien de Combinatoire, 78B.20 (2017), 12 pp.

# Steven N. Karp and Lauren K. Williams

# The *m*=1 Amplituhedron and Cyclic Hyperplane Arrangements

**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*. 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*. We deduce
that **A**_{n,k,1} is homeomorphic to a ball.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

The following versions are available: