Representation Theory, Combinatorics and Related Topics

表示论、组合学及相关主题

• 批准号: 14540041
• 负责人: KOIKE Kazuhiko
• 金额: $ 2.3万
• 依托单位: AOYAMA GAKUIN UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540041/
• 关键词: Classical groups; Spin groups; Brauer centralizer alegebras; Young diagrams; Jackson integrals; Ramanujan; Macdonald identity; Weyl groups

For the irreducible representations of the double covering groups Spin(N) of the special orthogonal groups SO(N), Koike gave natural generalization of Weyl's construction developed in his famous book 'The Classical Groups'. In the classical cases, the centralizer algebras of tine classical groups in the tensor product spaces of the natural representation play crucial roles. Koike showed that similar arguments hold for the Spin groups and described the centralizer algebras of the Spin groups and picked up all the irreducible representations of the Spin groups. It made the reason clear why apparently complicated identities such as Ramanujan's sum formulas etc hold and showed that the symmetries under the Weyl groups (or the root systems) are essential for them to hold. From the above view point, he gave consistent explanations of several identities which are found independently and further he found new identities, which gives the infinite product formulas of Jackson integral for root system F 4.

对于特殊正交群SO（N）的双覆盖群Spin（N）的不可约表示，小池给出了Weyl在他的著名著作《经典群》中所发展的构造的自然推广。在经典情形下，自然表示的张量积空间中的经典群的中心化子代数起着关键作用。小池表明，类似的论点举行的自旋组和描述的中心化代数的自旋组和拿起所有的不可约表示的自旋组。它的原因清楚地表明，为什么显然复杂的身份，如拉马努金的总和公式等举行，并表明，对称性下的外尔群（或根系统）是必不可少的，他们举行。从这一观点出发，他对几个独立发现的恒等式给出了一致的解释，并进一步发现了新的恒等式，给出了根系F4的杰克逊积分的无穷乘积公式。

项目成果

伊藤 雅彦: "アスキーウィルソン型積分とワイル群対称性を持つジャクソン積分との関係について"京都大学数理解析研究所講究録. 1296. 9-15 (2002)

伊藤雅彦：“关于 ASCII-Wilson 积分和具有 Weyl 群对称性的 Jackson 积分之间的关​​系”京都大学数学科学研究所 Kokyuroku。1296. 9-15 (2002)。

川村 友美: "Relations among the lowest degree of the Jones polynomial and geometric invariants for a closed positive braid"Comment.Math.Helv. 77. 125-132 (2002)

Tomomi Kawamura：“闭合正辫子的琼斯多项式的最低次数和几何不变量之间的关系”Comment.Math.Helv.77. 125-132 (2002)

Masahiko Ito: "Symmetry classification for Jackson integrals associated with the root system BCn"Compositio Mathematica. Vol.136. 209-216 (2003)

Masahiko Ito：“与根系统 BCn 相关的 Jackson 积分的对称性分类”Compositio Mathematica。

小池 和彦: "Spin Representations and Centralizer Algebras For Spin(2n+1)"京都大学数理解析研究所講究録. 1262. 9-29 (2002)

Kazuhiko Koike：“自旋表示和集中化代数（2n+1）”京都大学数学科学研究所 Kokyuroku。 1262. 9-29 (2002)

Masahiko Ito: "Symmetry classification for Jackson integrals associated with the root system BC_n"Compositio Mathematica. (accepted). (2003)

Masahiko Ito：“与根系统 BC_n 相关的杰克逊积分的对称分类”Compositio Mathematica。