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Hokkaido University Preprint Series in Mathematics >
Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem
| 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 |
| Publisher: | Department of Mathematics, Hokkaido University |
| 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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Submitter: 数学紀要登録作業用
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