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 $s,t\geq 0$, 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.