Bijections for faces of braid-type arrangements

Abstract

We establish a general bijective framework for encoding faces of some classical hyperplane arrangements. Precisely, we consider hyperplane arrangements in ${\mathbb{R}}^n$ whose hyperplanes are all of the form $\{x_i-x_j=s\}$ for some $i,j\in [n]$ and $s\in \mathbb{Z}$. Such an arrangement $A$ is \emph{strongly transitive} if it satisfies the following condition: if $\{x_i-x_j=s\}\notin A$ and $\{x_j-x_k=t\}\notin A$ for some $i,j,k\in [n]$ and $s,t\ge 0$, then $\{x_i-x_k=s+t\}\notin A$. For any strongly transitive arrangement $A$, we establish a bijection between the faces of $A$ and some set of decorated plane trees.