Chambers of arrangements of hyperplanes and Arrow's impossibility theorem

Abstract

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 ϵ H + : Ch → B by ϵ H + ( C ) = + (if C ⊆ H + ) and ϵ H + ( C ) = − (if C ⊆ H − ). Define ϵ H − = − ϵ H + . Let Ch m = Ch × Ch × ⋯ × Ch ( m times). Then the maps ϵ H ± induce the maps ϵ H ± : Ch m → B m . We will study the admissible maps Φ : Ch m → Ch which are compatible with every ϵ 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.