Dihedral sign patterns in $\mathcal{M}_{0,n}$

TL;DR

This paper studies the structure of the real moduli space _{0,n}, establishing a bijection between its connected components and dihedral orderings, and proves a conjecture relating sign patterns to extended u-relations.

Contribution

It constructs monomial maps between components of _{0,n} and proves a conjecture linking sign patterns with dihedral embeddings.

Findings

Connected components correspond to dihedral orderings.

Constructed monomial maps between components.

Proved a conjecture relating sign patterns and extended u-relations.

Abstract

The connected components of $\mathcal{M}_{0,n}(\mathbb{R})$ are in bijection with the $(n-1)!/2$ dihedral orderings of $[n]$. They are all isomorphic. We construct monomial maps between them, and use these maps to prove a conjecture of Arkani-Hamed, He, and Lam in the case of $\mathcal{M}_{0,n}$. Namely, we provide a bijection between connected components and sign patterns that are consistent with the extended $u$-relations for the dihedral embedding.