Combinatorics and Representation Theory of Nonlinear Differential Equations
Project/Area Number 
17540026

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Okayama University 
Principal Investigator 
YAMADA Hirofumi Graduate School of Science and Technology, Professor, 大学院自然科学研究科, 教授 (40192794)

CoInvestigator(Kenkyūbuntansha) 
YOSHINO Yuji Graduate School of Science and Technology, Professor, 大学院自然科学研究科, 教授 (00135302)
NAKAMURA Hiroaki Graduate School of Science and Technology, Professor, 大学院自然科学研究科, 教授 (60217883)
HIRANO Yasuyuki Naruto University of Education, Faculty of Education, Professor, 教育学部, 教授 (90144732)
TANAKA Katsumi Admission Center, Associate Professor, アドミッションセンター, 助教授 (60207082)
IKEDA Takeshi Okayama University of Science Faculty of Science, Senior Assistant Professor, 理学部, 講師 (40309539)

Project Period (FY) 
2005 – 2006

Project Status 
Completed (Fiscal Year 2006)

Budget Amount *help 
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2006: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2005: ¥1,700,000 (Direct Cost: ¥1,700,000)

Keywords  Schur functions / Affine Lie algebras / Symmetric groups / リリトン方程式 
Research Abstract 
I focused on the applications of the representation theory of the symmetric groups to certain nonlinear systems of differential equations. More precisely I investigated the Cartan matrices of the symmetric groups which play an important role in modular representation theory. It has been known that the coefficients of Qfunctions appearing in the expansion of 2reduced Schur functions are nonnegative integers. These are called the Stembridge coefficients. I noticed that the matrices of Stembridge coefficients are "similar" to the decomposition matrices for the 2modular representations of the symmetric groups. I proved that they are transformed to each other by simple column operations, and that the elementary divisors of the Cartan matrices and those of the socalled "Gartan matrices" coincide. Next I introduced the "compound basis" for the space of the symmetric functions and expanded (nonreduced) Schur functions in terms of our new basis. I found that the appearing coefficients are all integers. This compound basis arose naturally, at least for me, from representation theory of certain affine Lie algebras, which I have been studying for many years. At the present moment our basis is obtained only for the case of characteristic 2, but it is plausible that this exists for any characteristic p. A natural problem occurs: What is the transition matrix between the two bases, i.e., Schur function basis and our compound basis ? In a joint work with Mizukawa and Aokage, it is proved that the determinant of this transition matrix is a power of 2. This is a nontrivial fact.

Report
(3 results)
Research Products
(5 results)