2010 Fiscal Year Final Research Report
Recent development of special functions from the viewpoint of representation theory and integrals of complex variables
| Project/Area Number |
19340004
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Tokyo Institute of Technology |
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Principal Investigator |
MIMACHI Katsuhisa Tokyo Institute of Technology, 大学院・理工学研究科, 教授 (40211594)
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| Co-Investigator(Renkei-kenkyūsha) |
YOSHIDA Masaaki 九州大学, 大学院・数理学研究院, 教授 (30030787)
KUROKAWA Nobushige 東京工業大学, 大学院・理工学研究科, 教授 (70114866)
TAKATA Toshie 九州大学, 大学院・数理学研究院, 准教授 (40253398)
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| Project Period (FY) |
2007 – 2010
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| Keywords | ねじれホモロジー / 積分表示 / 接続問題 / 超幾何函数 / 交叉数 / モノドロミー / ジョーンズ多項式 / ヘッケ代数 |
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| Research Abstract |
We solve a connection problem associated with an integral of Selberg type, which appears in conformal field theory as a conformal block. The coefficients are expressed by the q-Racah polynomials, which consist of a family of orthogonal polynomials. On the other hand, we find a general framework that the connection problem is reduced to determining the intersection numbers of the associated twisted cycles. Indeed, we use this framework to solve the problem in the case of the generalized hypergeometric function. Moreover, we give a definition of the Jones polynomials in terms of intersection numbers of twisted cycles and derive explicit expressions of the Jones polynomials for some knots.
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