2007 Fiscal Year Final Research Report Summary
Representation theory of algebraic groups, Hecke algebras and canpex refiecion groups
| Project/Area Number |
17340003
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Nagoya University |
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Principal Investigator |
SHOJI Toshiaki Nagoya University, Graduate School of Matenmatics, Professor (40120191)
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| Co-Investigator(Kenkyū-buntansha) |
KAWANAKA Noriaki Osaka University, Graduate School of Information Science and Technology, Professor (10028219)
SHINODA Ken-ichi Sophia University, Faculty of Science and Technology, Professor (20053712)
GYOJA Akihiko Sophia University, Graduate School of Mathematics, Professor (50116026)
OKADA Soichi Sophia University, Gradate School of Mathematics, Professor (20224016)
ARIKI Susumu Kyoto University, Research lnstitute for Mathematics Science, Associate Professor (40212641)
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| Project Period (FY) |
2005 – 2007
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| Keywords | algebraic groups / Hecke algebras / complex reflection groups / cyclotomic q-Scqur algebras / Fock space / cellular aloebras / finite reductive groups / finite symmetric space |
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| Research Abstract |
We have studied on the following three themes.
1. Determination of scalars involved in Lusztig's conjecture concerning irreducible characters of finite reductive groups_G (F_q), and the determination of unipotent elements with certain good properties related to the algorithm of computing irreducible characters. In particular concerning the second problem, in the case where G (F_q) = SL_n(F_q), Sp_{2n}(F_q), SO_{2n+1} (F_q), SO {2n} (F_q) we have determined a class of good unipotent elements. In the case of Classical groups, this results holds also for the case where the characteristic is equal to 2. By this result, an algorithm computing Green functions of classical groups of even characteristic was established, which had involved certain ambiguity before.
2. It is known that Green functions of finite reductive groups is a polynomial in q. We have proved a formula for the values of Green functions obtained by substituting a root of unity for q. This type of formula was known in the case
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where G(F_q) = GL_n(F_q) by a combinatorial method. In this study, we have proved it in the general case by making use of induction theorem for Springer representations due to Lusztig.
3. We have studied the modular representation theory of Ariki-Koike algebras which are Hecke algebras associated to the complex reflection groups G(r,1,n), and the cyclotomic q-Schur algebras related to the Ariki-Koike algebras. We have constructed various subalgebras of cyclotomic q-Schur algebras and their quotients, and proved a product formulas for certain type of decomposition numbers, by comparing the decomposition numbers of them.
On the other hand, thanks to Yvonne's conjecture, it is conjectured that the decomposition numbers of cyclotomic q-Schur algebras are obtained from the transition matrix between standard bases and canonical bases of higher level Fock space. Based on this conjecture", we have proved a product formula for the Fock space which conjecturally corresponds to the original product formula Less
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