Combinatorial Representation Theory of Affine Lie Alqebras and Symmetric Groups
Project/Area Number 
11640001

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  OKAYAMA UNIVERSITY (2000) Hokkaido University (1999) 
Principal Investigator 
YAMADA Hirofumi Faculty of Science, OKAYAMA UNIVERSITY, Professor, 理学部, 教授 (40192794)

CoInvestigator(Kenkyūbuntansha) 
田口 雄一郎 北海道大学, 大学院・理学研究科, 助教授 (90231399)
斎藤 睦 北海道大学, 大学院・理学研究科, 助教授 (70215565)
山下 博 北海道大学, 大学院・理学研究科, 助教授 (30192793)
中島 達洋 明海大学, 経済学部, 講師 (00286006)
渋川 陽一 北海道大学, 大学院・理学研究科, 助手 (90241299)

Project Period (FY) 
1999 – 2000

Project Status 
Completed (Fiscal Year 2000)

Budget Amount *help 
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)

Keywords  Affine Lie Alqebras / Symmetric Groups / Schur Functions / Young Diagrams / アフィンリー環 / 指標 
Research Abstract 
My first attempt was to describe the weight basis of the basic representations of several typical affine Lie algebras. In particular, for the simplest affine Lie algebra A^<(1)>_1, I considered two realizations of the basic representation and found that the modular version of the Schur functions and Schur's Qfunctions occur as weight basis, respectively. Analysing these two realizations, I found an interesting phenomenon for the elementary divisors of the spin decomposition matrices for the symmetric group. Namely the elemntary divisors of the spin decomposition matrices for prime 2 are all powers of 2. Though this fact actually can be proved by a general theory of modular representations, I could give a simple proof of this by utilizing representations of the affine Lie algebra A^<(1)>_1. Studying the zonal polynomials, which are a specialization of the Jack polynomials, I found an interesting fact in the character tables of the symmetric group. Later I recognizes that this fact had been found more than 50 years ago by Littlewood, whose proof is a bit complicated. I gave a simple proof of this fact as well as its spin version with Hiroshi Mizukawa, a graduate student. The main tools for the proof are again Schur functions and Schur's Qfunctions. In the joint work with Takeshi Ikeda I could obtain all the homogeneous polynomial solutions for the nonlinear Schrodinger hierarchy. The schur functions indexed by the rectangular Young diagrams play an essential role in this theory.

Report
(3 results)
Research Products
(9 results)