Shelling the m=1 amplituhedron

Abstract

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. We show that $R_{n,l}$ is EL-shellable, resolving a problem posed by Karp and Williams. This gives a new proof that ${\mathcal{A}}_{n,k,1}$ is homeomorphic to a closed ball, which was originally proved by Karp and Williams. We also give explicit formulas for the $f$-vector and $h$-vector of $R_{n,l}$, and show that it is rank-log-concave and strongly Sperner. Finally, we consider a related poset $P_{n,l}$ introduced by Machacek (2019), consisting of all projective sign vectors of length $n$ with at most $l$ sign changes. We show that it is rank-log-concave, and conjecture that it is Sperner.

arXiv comment20 pages. v2: Minor changes

References in corpus (7)

• The Amplituhedron
• Total positivity, Grassmannians, and networks
• The totally nonnegative Grassmannian is a ball
• Regularity theorem for totally nonnegative flag varieties
• The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron
• The m=2 amplituhedron
• Boundary measurement and sign variation in real projective space

Cited by in corpus (1)

• Gradient flows, adjoint orbits, and the topology of totally nonnegative flag varieties