| Project/Area Number |
16K05067
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kagoshima University |
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Principal Investigator |
Minoru Itoh 鹿児島大学, 理工学域理学系, 教授 (60381141)
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| Co-Investigator(Kenkyū-buntansha) |
松本 詔 鹿児島大学, 理工学域理学系, 准教授 (60547553)
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| Project Period (FY) |
2016-04-01 – 2022-03-31
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| Project Status |
Completed (Fiscal Year 2021)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | 不変式論 / Cayley--Hamilton定理 / Schur多項式 / 1の原始冪根 / 行列式 / 正標数 / 超行列式 / derangement / 外積代数 / twisted immanant / Cayley-Hamilton定理 / Pfaffian / マトロイド / wreath積 / 表現論 / 代数学 / 微分 |
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| Outline of Final Research Achievements |
I introduced algebraic structures to discuss the Cayley-Hamilton theorem of higher order given by Y. Agaoka.
M. Hidaka and I proved that that the Schur polynomials in all nth primitive roots of unity are 1, 0, or -1, if n has at most two distinct odd prime factors. Moreover, J. Shimoyoshi and I studied on the existence of zero coefficients in the powers of the determinant polynomial of order n. D. G. Glynn proved that the coefficients of the mth power of the determinant polynomial are all nonzero, if m = p - 1 with a prime p. We proved that the converse also holds, if n is 3 or more.
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| Academic Significance and Societal Importance of the Research Achievements |
高階のCayley-Hamilton定理は, 様々な多項式環における不変式論で役立つが,「高階の行列環」における不変式論を考えれば, より直接的に, このCayley-Hamilton型定理自体が生成元の関係式の記述そのものと見なせる. この自然でわかりやすい見方で整理することができた.
日高氏との共同研究成果は, 円分多項式の係数に関する有名な事実の一般化で, 注目すべき結果である. 下吉氏との共同研究成果も, 証明はごく初等的だが, 予想外の結果である. またこの研究の元になったGlynnの結果についても, 偏極作用素を用いた自然で易しい不変式論的な新証明を得ることができた.
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