Representation theory and combinatorics of classical groups, quantum groups and Hecke algebras

经典群、量子群和赫克代数的表示论和组合学

• 批准号: 09640012
• 负责人: TERADA Itaru
• 金额: $ 1.92万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 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-09640012/
• 关键词: combinatiorics; representation theory; classical groups; Young diagrams; tableaux; Robinson-Schensted correspondence; Brauer diagrams; nilpotent matrices; ブラウアー図形; 指標; Robinson-Schensted対応; 分岐則; 小行列式; Well表現; Young図形

In a previous project, we gave an interpretation of a Robinson-Schensted-type correspondence between updown tableaux and Brauer diagrams, first discovered by Stanley, using alternating bilinear forms, flags and nilpotent matrices. In the current project, we obtained a more thorough result, namely we also showed the irreducibility of the variety formed by all triples : a nilpotent matrix, a flag and a nondegenerate alternating bilinear form such that the latter two are infinitesimally fixed by the first. This gives a closer parallelism of our result with Steinberg's interpretation of the original Ronbinson-Schensted correspondence. Similar geometric interpretations of other Robinson-Schensted-type correspondences are yet to be intestigated.In persuing the construction of a set of tableaux and a Robinson-Schensted-type correspondence which would combinatorially describe the decomposition of the tensor powers of the Weil representation of sp (2n, C), we gave a basis of combinatorial treatment by extending the use of specialization homomorphisms to certain infinite sums of universal characters. We assisted T.Roby with obtaining results for the stable region where the tensor power is large relative to the rank, and also for the case of any power with rank 2, which he nearly completed. The method is yet to be generalized to the case of any power with any rank.Further obtained were : results on characters of the classical groups by K.Koike ; results concerning the minor summation formulas and rhombus tilings by S.Okada ; results on multiplicity-free and discrete decomposabilities of certain infinite-dimensional representations of reductive groups by T.Kobayashi.

在以前的一个项目中，我们给出了一个解释的罗宾逊-申斯特型对应关系之间的updown tableaux和布劳尔图，首先发现的斯坦利，使用交替双线性形式，标志和幂零矩阵。在当前的项目中，我们得到了一个更彻底的结果，即我们还证明了所有三元组形成的簇的不可约性：幂零矩阵，标志和非退化交替双线性形式，使得后两者被第一个无穷小地固定。这给出了一个更密切的平行我们的结果与斯坦伯格的解释原来的龙宾森-申斯特对应。对其他Robinson-Schensted型对应的类似几何解释还有待研究。在构造一组组合描述sp（2n，C）的Weil表示的张量幂分解的表和Robinson-Schensted型对应时，我们通过将特殊化同态的应用推广到某些泛特征的无限和，给出了组合处理的基础。我们协助T.Roby获得了张量幂相对于秩较大的稳定区域的结果，以及秩为2的任何幂的情况，他几乎完成了。进一步得到的结果有：K.Koike关于经典群的特征标的结果; S.Okada关于次求和公式和菱形平铺的结果; T. Kobayashi关于约化群的某些无限维表示的无重性和离散可分解性的结果。

项目成果

小池和彦: "Qn representation of the classical groups" Amer.Math.Soc.Transl.Ser.2. 183. 79-100 (1998)

Kazuhiko Koike：“经典群的 Qn 表示”Amer.Math.Soc.Transl.Ser.2。183. 79-100 (1998)

小池和彦: "Representations of spinor groups and difference characters of SO(2n)" Adv.Math.128. 40-81 (1997)

Kazuhiko Koike：“SO(2n) 的旋量群和差分特征的表示”Adv.Math.128 (1997)。

Kazuhiko Koike: "On representation of the classical groups" Amer.Math.Soc.Transl.Ser.2183. 79-100 (1998)

小池和彦：“论经典群的表示”Amer.Math.Soc.Transl.Ser.2183。

岡田総一: "Applications of a minor-summation formulas to rectangular-shaped representations of classical groups" J.Aigebra. (発表予定).

Soichi Okada：“小求和公式在经典群的矩形表示中的应用”J.Aigebra（待提交）。

岡田総一・Christian Krattenthaler: "The number of rhombustilings of a "punctured"hexagon and the minor summation formula" Advances in Applied Mathematics. (発表予定).

Soichi Okada 和 Christian Krattenthaler：““穿孔”六边形的菱形数量和小求和公式”应用数学进展（待提交）。