2004 Fiscal Year Final Research Report Summary
Combinatorial Representation Theory which Center of Schur Functions
| Project/Area Number |
15540030
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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 |
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Principal Investigator |
YAMADA Hirofumi OKAYAMA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (40192794)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHINO Yuji OKAYAMA UNIVERSITY, Faculty of science, Professor, 理学部, 教授 (00135302)
NAKAMURA Hiroaki OKAYAMA UNIVERSITY, Faculty of science, Professro, 理学部, 教授 (60217883)
HIRANO Yasuyuki OKAYAMA UNIVERSITY, Faculty of science, Associate Professor, 理学部, 助教授 (90144732)
TANAKA Katsumi OKAYAMA UNIVERSITY, Faculty of science, Associate Professor, 理学部, 助教授 (60207082)
IKEDA Takeshi Okayama University of Science, Faculty of science, Lecturer, 理学部, 講師 (40309539)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Schur functions / Symmetric groups / affine Lie algebras |
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| Research Abstract |
This is an effort for understanding the role of Schur functions and Schur's Q functions, the projective analogue of Schur functions, in representation theory. To be more precise, we proved the following theorem. Schur functions associated with the rectangular Young diagrams occur as weight vectors of the basic representation of the affine Lie algebra of type D^{(2)}_2. And also, they turn out to be the homogeneous tau functions of the nonlinear Schroedinger hierarchy. The key idea for proving the above is to write down the representation spaces and operators in terms of fermions, and derive polynomials via the boson-fermion correspondence. We have succeeded in verifying the similar phenomena for the case of the affine Lie algebra of type A^{(2)}_2. In 2004 we considered the following problem. Clarify the nature of the coefficients in the 2 reduced Schur functions when expanded in terms of Schur's Q functions. Through some experimental computations in small rank cases, I had been convinced that these coefficients are of great interests, both from representation theoretical and combinatorial points of view. Finally we realized that these coefficients are nothing but the so-called Stembridge numbers. As a result we could relate these numbers with the representation theory of affine Lie algebras. Looking carefully at the table of these numbers, we found a simple formula for the elementary divisors of the Cartan matrices of the symmetric groups. More than 10 years ago, Olsson in Copenhagen gave a formula for those, which is expressed in terms of a generating function and is rather complicated. Our version is more direct and combinatorial.
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Research Products
(4 results)
