Bijections for hyperplane arrangements of Coxeter type

This talk is about real hyperplane arrangements whose hyperplanes are of the form {xi −xj = s} or {xi +xj = s}. We describe a bijective framework for a large family of such arrangements which we call transitive. For each transitive arrangement A, we give a bijection between the regions of A and a set of decorated trees. Particular cases include the families of Catalan, Shi, semiorder and Linial arrangements in type A, B, C, D and BC. We also derive some general enumerative formulas for such families of transitive arrangements.