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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:  24Aug2006 
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 halfspace $H^{+} $ and the other halfspace $H^{}$. 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

Submitter: 数学紀要登録作業用
