2001 Fiscal Year Final Research Report Summary
Classification problem of hypergeometric differential systems
| Project/Area Number |
12640149
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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 |
Basic analysis
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| Research Institution | HOKKAIDO UNIVERSITY |
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Principal Investigator |
SAITO Mutsumi Hokkaido Univ., Grad. School of Sci., Ass. Prof., 大学院・理学研究科, 助教授 (70215565)
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| Co-Investigator(Kenkyū-buntansha) |
SHIBUKAWA Youichi Hokkaido Univ., Grad. School of Sci., Instr., 大学院・理学研究科, 助手 (90241299)
MATSUMOTO Keiji Hokkaido Univ., Grad. School of Sci., Ass. Prof., 大学院・理学研究科, 助教授 (30229546)
YAMASHITA Hiroshi Hokkaido Univ., Grad. School of Sci., Ass. Prof., 大学院・理学研究科, 助教授 (30192793)
WACHI Akihito Hokkaido Inst. of Tech., Fac. of Gen. Edu., Lecturer, 総合教育研究部, 講師 (30337018)
YAMADA Hiro-fumi Okayama Univ., Faculty of Sci., Prof., 理学部, 教授 (40192794)
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| Project Period (FY) |
2000 – 2001
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| Keywords | hypergeometric / toric variety / Lie algebra / classification / dimension formula |
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| Research Abstract |
With support of many examples with a computer, and by communication with world-wide experts in several fields, we obtained the following results.
Mutsumi Saito generalized the classification theorem of A-hypergeometric systems to the cases when A is inhomogeneous and/or when we work in the analytic category. He also gave a dimension formula for the log-free series solutions when A is homogeneous, and a rank formula and the proof of the equivalence of Cohen-Macaulayness with the condition that the ranks are the same at all parameters, when A is homogeneous, and the convex hull of A is a simplex.
Hiroshi Yamashita obtained some results useful to know when an isotropy representation is irreducible. Furthermore he systematically constructed nonzero quotient representations of isotropy representations attached to discrete series.
Keiji Matsumoto clarified a pairing between twisted cohomology groups associated with generalized Airy functions. Writting a base of twisted cohomology groups by Young diagrams, he showed that for the base, the pairing can be explicitly written by skew-Schur polynomials.
Youichi Shibukawa solved the classification problem for R operators.
For the simplest affine Lie algebra A_1^<(1)> , using two of its realizations, Hiro-Fumi Yamada discovered the weight vectors are written by a modular version of Schur functions and Schur's Q-functions respectively.
Akihito Wachi has studied the structure of generalized Verma modules, in particular, their irreducibility, emphasizing their relations with invariant functions.
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