2007 Fiscal Year Final Research Report Summary
Representation theory of algebraic groups, Hecke algebras and canpex refiecion groups
Project/Area Number 
17340003

Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Nagoya University 
Principal Investigator 
SHOJI Toshiaki Nagoya University, Graduate School of Matenmatics, Professor (40120191)

CoInvestigator(Kenkyūbuntansha) 
KAWANAKA Noriaki Osaka University, Graduate School of Information Science and Technology, Professor (10028219)
SHINODA Kenichi 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)

Project Period (FY) 
2005 – 2007

Keywords  algebraic groups / Hecke algebras / complex reflection groups / cyclotomic qScqur algebras / Fock space / cellular aloebras / finite reductive groups / finite symmetric space 
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 ArikiKoike algebras which are Hecke algebras associated to the complex reflection groups G(r,1,n), and the cyclotomic qSchur algebras related to the ArikiKoike algebras. We have constructed various subalgebras of cyclotomic qSchur 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 qSchur 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

Research Products
(9 results)