Chambers of Arrangements of Hyperplanes and Arrow’s Impossibility Theorem

Let A be a nonempty real central arrangement of hyperplanes and Ch be the set of chambers of A. Each hyperplane H defines a half-space H and the other half-space H. Let B = {+,−}. For H ∈ A, define a map ǫ H : Ch → B by ǫ H (C) = + (if C ⊆ H) and ǫ H (C) = − (if C ⊆ H). Define ǫ H = −ǫ H . Let Ch = Ch×Ch× · · · ×Ch (m times). Then the maps ǫ H induce the maps ǫ H : Ch → B. We will study the admissible maps Φ : Ch → Ch which are compatible with every ǫ H . Suppose |A| ≥ 3 and m ≥ 2. Then we will show that A is indecomposable if and only if every admissible map is a projection to a component. When A is a braid arrangement, which is indecomposable, this result is equivalent to Arrow’s impossibility theorem in economics. We also determine the set of admissible maps explicitly for every nonempty real central arrangement.