Shelling the m=1 amplituhedron

TL;DR

This paper proves that the face poset of the m=1 amplituhedron is EL-shellable, providing a new proof of its topological properties and deriving combinatorial formulas, thus advancing understanding of its geometric and combinatorial structure.

Contribution

It establishes EL-shellability of the face poset of the m=1 amplituhedron and provides explicit formulas for its combinatorial invariants, resolving a prior open problem.

Findings

Proved the face poset is EL-shellable.

Derived explicit formulas for f- and h-vectors.

Showed the poset is rank-log-concave and strongly Sperner.

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.