2023 Fiscal Year Final Research Report
Fusion of algebra, geometry and combinatorics based on the roots of Poincare polynomials of hyperplane arrangements
| Project/Area Number |
20K20880
|
|---|
| Research Category |
Grant-in-Aid for Challenging Research (Exploratory)
|
| Allocation Type | Multi-year Fund |
|---|
| Review Section |
Medium-sized Section 11:Algebra, geometry, and related fields
|
|---|
| Research Institution | Rikkyo University (2023)
Kyushu University (2020-2022) |
|---|
Principal Investigator |
ABE TAKURO 立教大学, 理学部, 教授 (50435971)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
沼田 泰英 北海道大学, 理学研究員, 教授 (00455685)
鍛冶 静雄 九州大学, マス・フォア・インダストリ研究所, 教授 (00509656)
|
|---|
| Project Period (FY) |
2020-07-30 – 2024-03-31
|
| Keywords | 超平面配置 / ポアンカレ多項式 / 整数根 / 対数的ベクトル場 / オイラー制限射 / グラフ配置 / 局所化と制限 |
|---|
| Outline of Final Research Achievements |
In this research project we study the meaning of the integer roots of Poincare polynomials of hyperplane arrangements. They are determined combinatorially, and when the arrangement is free they have a good algebraic meaning by Terao's factorization theorem. However, in general there are no good way to understand them. We tried to understand them, and obtained two main results. The first one is on the relation of integer roots of the original arrangement and its localization/restriction. Localizations and restrictions are important operations in the research of arrangements, but how their integer roots are related have not been well-understood. By using the surjectivity of the Euler restriction map, we related them. Second, we studied integer roots of arrangements coming from digraphs. There are two important classes of such arrangements called Shi and Ish arrangements. We found a graphic operation keeping the integer roots (in fact, freeness is kept) and connect these arrangements.
|
| Free Research Field |
超平面配置の数学、代数学
|
| Academic Significance and Societal Importance of the Research Achievements |
超平面配置とはベクトル空間中の有限個の超平面の集合であり、最も簡単な例は平面に有限本の直線を描いた図形である。これは植木算の高次元化といえ、植木算における区間の数え上げに対応するものが、超平面の交わりの情報いわゆる交差格子とそこから得られる区間の一般化であるポアンカレ多項式である。このポアンカレ多項式が整数根を持つ場合があるが、その意味は自由配置の場合以外謎に包まれていた。本研究ではその根の意味に切り込むことで、超平面配置理論の深い理解を目指し、特に超平面配置を制限したりある点の周りのローカルな情報に集中した場合の挙動、及びグラフから構成される配置の整数根について研究を深めることができた。
|
