2003 Fiscal Year Final Research Report Summary
Rationality of elgenvalues of Cartan matrices in finite groups
| Project/Area Number |
14540012
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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 Agriculture And Technology |
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Principal Investigator |
WADA Tomoyuki Tokyo University Agriculture And Technology, Faculty of Technology, Professor, 工学部, 教授 (30134795)
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| Co-Investigator(Kenkyū-buntansha) |
ENOMOTO Yoko (USAMI Yoko) Ochanomizu Woman's University, Science, Professor, 理学部, 教授 (90151993)
KIYOTA Masao Tokyo Medical and Dental University, General Education, Professor, 教養部, 教授 (50214911)
YAMAGATA Kunio Tokyo University Agriculture And Technology, Faculty of Technology, Professor, 工学部, 教授 (60015849)
FUKUSHIMA Hiroshi Gunma University, Education, Professor, 教育学部, 教授 (30125869)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Finite group / Modular representation / Block / Cartan matrix / Frobenius-Perron eigenvalue / Elementary divisor / Morita equivalence / Brauer correspondence |
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| Research Abstract |
Let G be a finite group and Fbe an algebraically closed field of characteristic p>0. Let FG be the group algebra and let B be a block of FG with defect group D. Let C_B be the Cartan matrix of Band let ρ (B)be the Frobenius-Perron eigenvalue of C_B. It is the purpose of this research to investigate when the sets of eigenvalues and elementary divisors of C_B coincide. We have the following results and conjecture.
1.Bis of finite type (i.e.Dis cyclic), or of tame type (i.e.p=2 and Dis dihedral, generalized quaternion or semidihedral), then the following are equivalent. If G is p-solvable, then The following (1) and (2) are equivalent.
(1)The sets of eigenvalues and elementary divisors of C_B coincide.
(2)ρ(B)=|D|.
(3)ρ(B)is an integer.
Furthermore, in this case, B and its Brauer correspondent b are Morita equivalent, if B is of finite type or tame type.
2.Conjecture Let E be the set of elementary divisors of C_B and let f_B(x) be the characteristic polynomial of C_B. Let f_B(x)=f_1(x)f_2(x)・・・f_r(x) be the Z-irreducible decomposition. Let R_i be the set of roots of f_i(x). Then there is a disjoint decomposion E=E_1 U・・・UE_r such that the following (i),(ii),(iii) hold.
(i)|R_i|= |E_i| for all i.
(ii)The product of eigenvalues and elementary divisors in R_i and E_i coinside.
(iii)Let ρ(B) is in R_1. Then |D| is in E_1.
3.We have the following affirmative result for Conjecture. If B is of finite type and the number of irreducible B-modules≦5,or B is of tame type then Conjecture is true. Furthermore, in this case, if ρ(B,)is a root of f_1(x), then deg f_1≧deg f_i for all i. Moreover, we have verified Conjecture being true for non abelian finite simple groups whose Cartan matrices are known.
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