2004 Fiscal Year Final Research Report Summary
Research on arrangements of hyperplane
| Project/Area Number |
14340018
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tokyo Metropolitan University |
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Principal Investigator |
THRAO Hiroaki Tokyo Metropolitan University, Department of Math, Professor, 理学研究科, 教授 (90119058)
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| Co-Investigator(Kenkyū-buntansha) |
OKA Mutsuo Tokyo Metropolitan University, Department of Math., Professor, 理学研究科, 教授 (40011697)
MARTIN Guest Tokyo Metropolitan University, Department of Math., Professor, 理学研究科, 教授 (10295470)
TOKUNAGA Hiroo Tokyo Metropolitan University, Department of Math., Assistant Professor, 理学研究科, 助教授 (30211395)
NAKASHIMA Toru Tokyo Metropolitan University, Department of Math., Assistant Professor, 理学研究科, 助教授 (20244410)
NAKAMULA Ken Tokyo Metropolitan University, Department of Math, Professor, 理学研究科, 教授 (80110849)
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| Project Period (FY) |
2002 – 2004
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| Keywords | hyperplane arrangement / hypergeometric integral / local system cohomology / reflection group |
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| Research Abstract |
1. In "Moduli space of combinatorially equivalent arrangements of hyperplanes and logarithmic Gauss-Manin connections."(H.Terao), we scrutinized the structure of the boundary divisor of the moduli space of hyperplane arrangements with a fixed number of hyperplanes and studied the Gauss-Manin connection matrix of a local system on each hyperplane arrangement. Especially we proved that every pole is logarithmic.
2. In "Algebras generated by reciprocals of linear forms."(H.Terao), we obtained the explicit formula of the Poincare polynomial of the graded algebra generated by reciprocals of linear forms over a field.
3. In "Multiderivations of Coxeter arrangements."(H.Terao), we showed that the module of vector fields tangent to a Coxeter arrangement with a multiple degree is a free module over a polynomial ring. Furthermore, an explicit basis was obtained.
4. The paper "The Poincard series of the algebra of rational functions which are regular outside hyperplanes."(H.Horiuchi, H.Terao),is a s
… More
equel to 2. It studies the algebra of rational functions with poles along a hyperplane arrangement. The two-variable Poincare series of the algebra was explicitly determined when a bi-degree is introduced to the algebra.
5. In "Bases of the contact-order filtration of derivations of Coxeter arrangements."(H.Terao), the finite Coxeter arrangement which is a collection of mirrors of reflections of a classical of Coxeter groups. Especially a basis was constructed in a geometric way. Moreover the relationship to the basis constructed by M. Yoshinaga(RIMS) in 2002.
6. The proof of the Edelman-Reiner conjecture by M. Yoshinaga is a marvelous application of the theory of free arrangements and the factorization theorem (H. Terao,1981). In "On the proof of the Edelman-Reiner conjecture" (in Japanese), we gave an overview of the proof and discussed the possible future research.
7. Among various applications of hyperplane arrangements, an application to statistics is actively studied. More concretely, in the ranking theory, which enables us to avoid the Impossibility Theorem by Kenneth Arrow in the social choice theory, a problem concerning the number of possible rankings can be solved by counting the number of chambers in "Hyperplane arrangements and ranking - a contact of mathematics to social science - "(in Japanese). A rigorous proof is found in "Ranking Patterns of the Unfolding Model and Arrangements "(H. Kamiya, R Orlik, A. Takemura, H. Terao). Less
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