Shelling the m=1 amplituhedron

Understand

The amplituhedron $\mathcal{A}_{n,k,m}$ was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory.

• It is a projection inside the Grassmannian $\text{Gr}_{k,k+m}$ of the totally nonnegative part of $\text{Gr}_{k,n}$.
• Karp and Williams (2019) studied the $m=1$ amplituhedron $\mathcal{A}_{n,k,1}$, giving a regular CW decomposition of it.
• Its face poset $R_{n,l}$ (with $l := n-k-1$) consists of all projective sign vectors of length $n$ with exactly $l$ sign changes.