2011 Fiscal Year Final Research Report
Eigenvalues of Cartan matrices and Morita equivalences of blocks in finite groups
| Project/Area Number |
21540009
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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 | Tokyo University of Agriculture and Technology |
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Principal Investigator |
WADA Tomoyuki 東京農工大学, 大学院・工学研究院, 教授 (40003008)
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| Co-Investigator(Renkei-kenkyūsha) |
YAMAGATA Kunio 東京農工大学, 大学院・工学研究院, 教授 (60015849)
KIYOTA Masao 東京医科歯科大学, 教養部, 教授 (50214911)
FUKUSHIMA Hiroshi 群馬大学, 教育学部, 教授 (30125869)
KUNUGI Naoko 東京理科大学, 理学部, 准教授 (50362306)
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| Project Period (FY) |
2009 – 2011
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| Keywords | 群の表現論 |
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| Research Abstract |
Let G be a finite group and H be a normal subgroup of G with p'-index. Let B be a p-block of G and b be any p-block of H covered by B. Then in Theorem 1. 1 of the work of Okuyama-Wada[ Okuyama-Wada, Contemp. Math., 524, 2010], we have proved that the largest eigenvalues of B and b are equal. Furthermore, we have proved that if B satisfies a certain condition(#) on the degrees of irreducible Brauer characters in B, then there exists an eigenvalueλof B such that the(π)-part ofλis equal to the order of defect group D of B. Any p-block of p-solvable group satisfies(#), but it does not hold in general. There exist counter examples for p> 3, however we could not find a counter example for p=2. How is a 2-block of the symmetric group? Calculating the degrees of irreducible Brauer characters, any 2-block of the symmetric group seems to satisfy(#). Furthermore, Kiyota, Okuyama and Wada have recently proved a stronger result than(#) that any 2-block of the symmetric group of arbitrary degree has a unique irreducible Brauer character of height 0[ Kiyota-Okuyama-Wada, accepted]. This generalizes the theorem of Fong, James that the degree of every non-trivial irreducible 2-Brauer characters of the symmetric group is even. This theorem has never been known. We could not discover this remarkable fact if we would not consider eigenvalues of the Cartan matrices of finite groups.
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