Bijections for faces of braid-type arrangements

We establish a general bijective framework for encoding faces of some classical hyperplane arrangements. Precisely, we consider hyperplane arrangements in Rⁿ whose hyperplanes are all of the form ᵢ-xⱼ=s\ for some i,j∈[n] and s∈ Z. Such an arrangement A is strongly transitive if it satisfies the following condition: if ᵢ-xⱼ=s\∉ A and ⱼ-xₖ=t\∉ A for some i,j,k∈ [n] and s,t≥ 0, then ᵢ-xₖ=s+t\∉ A. For any strongly transitive arrangement A, we establish a bijection between the faces of A and some set of decorated plane trees.