2007 Fiscal Year Final Research Report Summary
Study on representation theory of the classical groups and their related combinatorics
| Project/Area Number |
18540046
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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 | Aoyama Gakuin University |
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Principal Investigator |
KOIKE Kazuhiko Aoyama Gakuin University, College of Science and Engineering, Professor (70146306)
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| Co-Investigator(Kenkyū-buntansha) |
ITO Masahiko Aoyama Gakuin University, College of Science and Engineering, Associate Professor (30348461)
TANIGUCHI Kenji Aoyama Gakuin University, College of Science and Engineering, Associate Professor (20306492)
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| Project Period (FY) |
2006 – 2007
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| Keywords | algebra / combinatoics / representation theor / classical groups / Weyl groups |
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| Research Abstract |
For the classical Lie groups, the famous formulas by Herman Weyl, which say that the denominators of their irreducible characters are completely decomposed into the product of the linear factors, are regarded as the determinants whose components are the irreducible characters of rank one Lie groups of the same type. From this view point, Koike and Ito generalize these formulas and give some formulas that the determinants, whose components (the row indices are parameterized by the Young diagrams included in some fixed rectangular and the column indices are parameterized by the subsets of the variables) are given by the irreducible characters corresponding to the Young diagrams of the row indices in their variables prescribed by the column indices, are completely decomposed into the products of the linear factors. The resulting formulas turn out to be the powers of the original denominator formulas. These formulas play an crucial role in the result by Aomoto-Ito, in which they construct the fundamental system of solutions of holonmic q-difference system of Jackson integrals of type BC_n. Ito also gives explicit connection formulas of the solutions of holonmic q-difference system of Jackson integrals of type BC_n. Taniguchi gives an elementary construction for the invariant polynomials of type F_4 under the action of its Weyl group, using the degree 2 invariant polynomials under the action of Weyl group of type D_4, based on the fact that Weyl group of type D(-4) is included in a normal subgroup of Weyl group of type F_4
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