Mathematics – Combinatorics
Scientific paper
2006-08-23
Advances in Math. 214 (2007), 366-378
Mathematics
Combinatorics
Scientific paper
10.1016/j.aim.2007.02.006
Let ${\mathcal A}$ be a nonempty real central arrangement of hyperplanes and ${\rm \bf Ch}$ be the set of chambers of ${\mathcal A}$. Each hyperplane $H$ defines a half-space $H^{+} $ and the other half-space $H^{-}$. 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 ${\rm \bf Ch}^{m} = {\rm \bf Ch}\times{\rm \bf Ch}\times...\times{\rm \bf Ch} (m\text{times}).$ 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 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.
No associations
LandOfFree
Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-339602