2000 Fiscal Year Final Research Report Summary
Combinatorial Representation Theory of Affine Lie Alqebras and Symmetric Groups
| Project/Area Number |
11640001
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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 | OKAYAMA UNIVERSITY (2000)
Hokkaido University (1999) |
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Principal Investigator |
YAMADA Hiro-fumi Faculty of Science, OKAYAMA UNIVERSITY, Professor, 理学部, 教授 (40192794)
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| Project Period (FY) |
1999 – 2000
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| Keywords | Affine Lie Alqebras / Symmetric Groups / Schur Functions / Young Diagrams |
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| 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 Q-functions 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 Q-functions.
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.
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