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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 →KAKEN DB |
| Keywords: | Arrangement of hyperplanes | | Chambers | | Braid arrangements | | Arrow's impossibility theorem |
| Issue Date: | 10-Sep-2007 |
| Publisher: | Elsevier Inc. |
| Journal Title: | Advances in Mathematics |
| Volume: | 214 |
| Issue: | 1 |
| Start Page: | 366 |
| End Page: | 378 |
| Publisher DOI: | 10.1016/j.aim.2007.02.006 |
| Abstract: | Let A be a nonempty real central arrangement of hyperplanes and Ch be the set of chambers of A. Each hyperplane H defines a half-space H+ and the other half-space H−. Let B={+,−}. For H ∈ A, define a map ∈+H : Ch → B by ∈+H(C) = + (if C ⊆ H+) and ∈+H(C) = - (if C ⊆ H−). Define ∈-H = -∈+H. Let Chm=Ch×Ch×･･･×Ch (m times). Then the maps ∈±H induce the maps ∈±H : Chm → Bm. We will study the admissible maps Φ : Chm → Ch which are compatible with every ∈±H. Suppose |A| ≥ 3 and m ≥ 2. Then we will show that A is indecomposable if and only if every admissible map is a projection to a component. When 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. |
| Relation: | http://www.sciencedirect.com/science/journal/00018708 |
| Type: | article (author version) |
| URI: | http://hdl.handle.net/2115/30120 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)
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Submitter: 寺尾 宏明
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