Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem

Abstract

Let $\mathcal{A}$ be a nonempty real central arrangement of hyperplanes and $\mathbf{Ch}$ be the set of chambers of $\mathcal{A}$. Each hyperplane $H$ defines a half-space $H^{+}$ and the other half-space $H^{-}$. Let $B=\{+,-\}$. For $H\in \mathcal{A}$, define a map ${\epsilon }_H^{+}:\mathbf{Ch}\to B$ by ${\epsilon }_H^{+}(C)=+\text{(if}C\subseteq H^{+})\text{and}{\epsilon }_H^{+}(C)=-\text{(if}C\subseteq H^{-}).$ Define ${\epsilon }_H^{-}=-{\epsilon }_H^{+}.$ Let ${\mathbf{Ch}}^m=\mathbf{Ch}\times \mathbf{Ch}\times ...\times \mathbf{Ch}(m\text{times}).$ Then the maps ${\epsilon }_H^{\pm }$ induce the maps ${\epsilon }_H^{\pm }:{\mathbf{Ch}}^m\to B^m$. We will study the admissible maps $\Phi :{\mathbf{Ch}}^m\to \mathbf{Ch}$ which are compatible with every ${\epsilon }_H^{\pm }$. Suppose $|\mathcal{A}|\ge 3$ and $m\ge 2$. Then we will show that $\mathcal{A}$ is indecomposable if and only if every admissible map is a projection to a omponent. When $\mathcal{A}$ 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.