| Project/Area Number |
19K14516
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Soka University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2022-03-31
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| Project Status |
Completed (Fiscal Year 2021)
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| Budget Amount *help |
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 代数群 / 代数幾何学 / 組み合わせ幾何学 / 数論 / 幾何学的不変式論 / 球面式建物 / algebraic groups / invariant theory / spherical buildings / representation theory |
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| Outline of Research at the Start |
We study Serre's notion of complete reducibility of subgroups of reductive algebraic groups (matrix groups) via geometric invariant theory (a branch of algebraic geometry) and the theory of spherical buildings (highly combinatorial objects).
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| Outline of Final Research Achievements |
In this project, we studied Serre's notion of complete reduciblity of subgroups of reductive algebraic groups using "geometric invariant theory" (a part of algebraic geometry) and "the theory of spherical buildings (highly symmetrical combinatorial objects). So far, the study of complete reducibility had been done by representation theoretic methods that were ad hoc, using different arguments depending on the "types" of reductive algebraic groups. In this research, I invented a unified method via geometric invariant theory and the theory of spherical buildings proving various results concerning complete reducibility in very short arguments.
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| Academic Significance and Societal Importance of the Research Achievements |
これまでタイプ別に分析されていた代数群の構造に幾何学的不変式論を用いた統一的理解とビルディングの理論を用いたトポロジー的解釈を与え、より構造を理解しやすくした。またこれまでほとんど分析されてこなかったより複雑な代数群のケース（体がperfectでないケース）に幾何学的不変式論を適用するとこれまで通りの結果が成り立つ場合と成り立たない場合がある事を具体例を使ってその原因とともに示した。この結果は特に数論への応用に対して重要であると考えられ、今後の発展がより期待される。
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