Deformations of Coxeter hyperplane arrangements

Abstract

We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement x_i - x_j = 1, 1 \leq i<j \leq n, is equal to the number of alternating trees on n+1 vertices. Remarkably, these numbers have several additional combinatorial interpretations in terms of binary trees, partially ordered sets, and tournaments. More generally, we give formulae for the number of regions and the Poincar'e polynomial of certain finite subarrangements of the affine Coxeter arrangement of type A_{n-1}. These formulae enable us to prove a "Riemann hypothesis" on the location of zeros of the Poincar'e polynomial. We also consider some generic deformations of Coxeter arrangements of type A_{n-1}.