1998 Fiscal Year Final Research Report Summary
Combinatorial aspects of representations of groups and algebras
| Project/Area Number |
09640001
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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 | Hokkaido University |
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Principal Investigator |
YAMADA Hirofumi Grad.School of Science, Hokkaido Univ., Asso.Prof., 大学院・理学研究科, 助教授 (40192794)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAJIMA Tatsuhiro Faculty of Economics, Meikai Univ., Lec., 経済学部, 講師 (00286006)
TERAO Hiroaki Grad.School of Science, Tokyo Metropolitan Univ., Prof., 大学院・理学研究科, 教授 (90119058)
SGIBUKAWA Youichi Grad.School of Science, Hokkaido Univ., Inst., 大学院・理学研究科, 助手 (90241299)
SAITO Mutsumi Grad.School of Science, Hokkaido Univ., Asso.Prof., 大学院・理学研究科, 助教授 (70215565)
YAMASHITA Hiroshi Grad.School of Science, Hokkaido Univ., Asso.Prof., 大学院・理学研究科, 助教授 (30192793)
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| Project Period (FY) |
1997 – 1998
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| Keywords | affine Lie algebras / Schur functions / decomposition matrices / complex reflection groups |
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| Research Abstract |
I focused on a relationship of Schur's Q-functions and affine Lie algebras. First I found that the Q-functions, expressed as polynomials of power sum symmetric functions, form a weight basis for the basic representation of certain affine Lie algebras, realized on a polynomial ring. Q-functions are parametrized by the strict partitions. Using some combinatorics of Young diagrams, I determined the weight of the given Q-function. This procedure was applied to the simplest affine lie algebra $A^{(1)}_1$ to find an identity satisfied by Schur functions and Q-functions indexed by some specific partitions. At first this identity seemed funny : However this was proved to be true by making use of decomposition matrices of the spin representations of the symmetric group. By virtue of this fact, I turned to a study of the decomposition matrices themselves. As a first result I proved that the determinant of the decomposition matrix of the spin representations is equal to a power of two when the characteristic equals two.
Another feature of my research is the so called "higher Specht polynomials" for the complex reflection group G(r, p, n). The group G(r, p, n) acts on the polynomial ring of n variables. The "coinvariant ring" is the quotient by the ideal which is generated by invariants over the group. It is known that the action of G(r, p, n) on this coinvariant ring is isomorphic to the regular representation. The higher Specht polynomials appear naturally as basis vectors of each irreducible component.
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