Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem
Preprint Series # 799 Terao, Hiroaki Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem. (2006); AbstractLet ${\mathcal A}$ be a nonempty real central arrangement of hyperplanes and ${\rm \bf Ch}$ be the set of chambers of ${\mathcal A}$. Each hyperplane $H$ makes a halfspace $H^{+} $ and the other halfspace $H^{}$. Let $B = \{+, \}$. For $H\in {\mathcal A}$, define a map $\epsilon_{H}^{+} : {\rm \bf Ch} \to B$ 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 ${\rm \bf Ch}^{m} = {\rm \bf Ch} \times{\rm \bf Ch}\times\dots\times{\rm \bf Ch} \,\,\,(m\text{~times}).$ Then the maps $\epsilon_{H}^{\pm}$ induce the maps $\epsilon_{H}^{\pm} : {\rm \bf Ch}^{m} \to B^{m} $. We will study the admissible maps $\Phi : {\rm \bf Ch}^{m} \to {\rm \bf Ch}$ which are compatible with every $\epsilon_{H}^{\pm}$. Suppose ${\mathcal A}\geq 3$ and $m\geq 2$. Then we will show that ${\mathcal A}$ is indecomposable if and only if every admissible map is a projection to a component. 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.
