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# Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem

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 Title: Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem Authors: Terao, Hiroaki Browse this author Keywords: arrangement of hyperplanes chambers braid arrangements Arrow's impossibility theorem Issue Date: 24-Aug-2006 Journal Title: Hokkaido University Preprint Series in Mathematics Volume: 799 Start Page: 1 End Page: 13 Abstract: 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 $H$ makes 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 ${\rm \bf Ch}^{m} = {\rm \bf Ch} \times{\rm \bf Ch}\times\dots\times{\rm \bf Ch} \,\,\,(m\text{_*_times}).$ 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. Type: bulletin (article) URI: http://hdl.handle.net/2115/69607 Appears in Collections: 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics

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