Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem
| dc.creator | Terao, Hiroaki | |
| dc.date | 2006-08-23 | |
| dc.date | 2007-02-10 | |
| dc.date.accessioned | 2026-07-07T08:14:03Z | |
| dc.date.available | 2026-07-07T08:14:03Z | |
| dc.description | 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 $ε_{H}^{+} : {\rm \bf Ch} \to B$ by $ε_{H}^{+} (C)=+ \text{(if} C\subseteq H^{+}) \text{and} ε_{H}^{+} (C)= - \text{(if} C\subseteq H^{-}).$ Define $ε_{H}^{-}=-ε_{H}^{+}.$ Let ${\rm \bf Ch}^{m} = {\rm \bf Ch}\times{\rm \bf Ch}\times...\times{\rm \bf Ch} (m\text{times}).$ Then the maps $ε_{H}^{\pm}$ induce the maps $ε_{H}^{\pm} : {\rm \bf Ch}^{m} \to B^{m} $. We will study the admissible maps $Φ: {\rm \bf Ch}^{m} \to {\rm \bf Ch}$ which are compatible with every $ε_{H}^{\pm}$. 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. | |
| dc.identifier | https://arxiv.org/abs/math/0608591 | |
| dc.identifier | http://arxiv.org/abs/math/0608591 | |
| dc.identifier | Advances in Math. 214 (2007), 366-378 | |
| dc.identifier | doi:10.1016/j.aim.2007.02.006 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/132988 | |
| dc.subject | Combinatorics | |
| dc.subject | 32S22;91B14 | |
| dc.title | Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem | |
| dc.type | text |