Many-Sided study of Selberg type integrals

Selberg 型积分的多方面研究

• 批准号: 09440064
• 负责人: KANEKO Jyoichi
• 金额: $ 3.52万
• 依托单位: KYUSHU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440064/
• 关键词: Selberg type integral; deformed Solberg type; twisted (co-) homology group; Gauss-Marin system; Jack polynomial; A-hypergeometric function; A-hypergeometric ideal; weakly 1-complete manifold; 変型Selberg型積分; Joak多項式; Coken-Macaulay環; weakly 1-cor

In our research, we mainly studied the twisted (co-)homology groups and the holonomic systems attached to the (deformed) Selberg type integrals. We regard the integral as the dual pairing of the twisted homology group and the twisted cohomology group, and so the fundamental problem is to construct the bases of these (co-)homology groups. We explicitly constructed the bases of the (co-)homology groups, and noticed that these bases are nothing but the ones obtained from the beta-nbc bases due to Falk and Terao. We also calculated the Gauss-Manin system explicitly in graph-theoretical terms (Duke Math. J.). Our de-formed Selberg type integral is also an example of A-type hypergeometric functions due to Gelfand-Kapranov-Zelevinsky. We have been investigating the so-called A-hypergeometric ideal describing the holonomic system of the integral, especially on its Cohen-Macaulay property and construction of the basis.The conjecture of Forrester predicts that the value of certain generalization of the original Selberg integral is also given by an explicit GAMMA product. We verified this in some cases by using the integration formula of Jack polynomials (Contemporary Math., to appear). We have been currently, working on this conjecture by employing Dunkl operators.Kazama (with S.Takayama) solved in the negative the long-standing conjecture of S.Nakano concerning **-problem on weakly 1-complete manifolds (Nagoya Math. J., to appear). They also investigated related problems on complex Lie groups (Nagoya Math. 3., to appear).

在我们的研究中，我们主要研究了（变形）Selberg类型积分附加的扭曲的（）同源组和自动系统。我们认为该积分是扭曲同源组和扭曲的共同体学组的双重配对，因此基本问题是构建这些（共同）同源组的基础。我们明确构建了（共同）同源组的基础，并注意到这些碱基不过是由于falk和terao而从beta-nbc碱中获得的基础。我们还用图理论术语明确地计算了高斯 - 曼宁系统（DukeMath。J.）。由于Gelfand-Kapranov-Zelevinsky，我们的De-formed Selberg型积分也是A型超几何函数的一个示例。我们一直在研究描述积分的全体性系统的所谓的A-HYPERMOTITRICTIOME，尤其是在其Cohen-Macaulay特性和基础上的构建方面。Forrester的猜想预测，明确的伽玛产品也给出了原始Selberg积分的某些概括的价值。在某些情况下，我们通过使用Jack多项式的集成公式（现代数学）来验证这一点。目前，我们一直在通过使用dunkl operators.kazama（与s.takayama）解决这一猜想的工作，以否定的方式解决了S.Nakano关于**的长期以来的猜想 - 出现在弱的1份歧管上（NagoyaMath。J.，出现）。他们还研究了有关复杂谎言组的相关问题（NagoyaMath。3。出现）。

项目成果

H.Kazama: "oo-problem on weakly 1-cowplete Kaikk mauibolds" Nagoya Moth.J.(to appear).

H.Kazama：“弱 1-cowplete Kaikk mauibolds 上的 oo 问题”Nagoya Moth.J.（出现）。

H.Kazama: "∂∂-problem on weakly 1-complete Kahler manifolds" Nagoya Math. J.to appear.

H.Kazama：“∂∂-弱 1-完全卡勒流形问题”Nagoya Math J. 出现。

H.Kazama: "Some remarks on conplex Lie growps" Nagoya Math.J.(to appear).

H.Kazama：“关于复数李生长的一些评论”Nagoya Math.J.（待发表）。

J.Kaneko: "Constant term ideutities of Forrester-Zeilberger-Cooper" Discrete Math.173. 79-90 (1997)

J.Kaneko：“Forrester-Zeilberger-Cooper 的常数项概念”离散数学.173。

J.Kaneko: "A_1Ψ_1 summation theorem for Macdonald polynouials" The Romanujau J.2. 397-386 (1998)

J.Kaneko：“麦克唐纳多项式的 A_1Ψ_1 求和定理”Romanujau J.2 (1998)。