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Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem

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Please use this identifier to cite or link to this item:http://doi.org/10.14943/83949

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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