2010 Fiscal Year Final Research Report
Artinian Gorenstein rings with the action of the symmetric group
| Project/Area Number |
19540052
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Tokai University |
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Principal Investigator |
WATANABE Junzo Tokai University, 理学部, 教授 (40022727)
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| Project Period (FY) |
2007 – 2010
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| Keywords | 完全交叉 / レフシェッツ性 / ヘシアン / 高次ヘシアン / ジョルダン標準形 / ゴレンスタイン環 / 完全交叉環 |
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| Research Abstract |
We let the symmetric group act on the polynomial ring, and used the Weyl duality to obtain certain results on the invariant coalgebras by Young subgroups of the symmetric group. As results, we obtained the following. (1) We introduced the notion of higher Hessians and using it, we characterized the strong Lefschetz elements in Artinian Gorenstein local rings. (2) We proved that the coalgebras of invariants by the Young subgroups have the strong Lefschetz property. (3) We obtained the irreducible decomposition of the exterior algebra of the differential module over the polynomial ring. (4) We showed some properties of power some symmetric functions in three variables and made a conjecture to the problem to ask in which choice of degrees the power sums form a regular sequence.
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