Restricting Specht Modules of Finite General Linear Groups to the Unitriangular Subgroup

将有限一般线性群的谱模限制为单位三角子群

• 批准号: 219367947
• 负责人: Professor Dr. Richard Dipper
• 金额: --
• 依托单位: Abteilung für Darstellungstheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/219367947?language=en
• 关键词: Restricting; Specht; Modules; Finite; General

One of the most important problems in the representation theory of finite groups G is to find the irreducible complex representations of G. For the finite general linear groups G = GLn(q) of invertible matrices of order n over the finite field Fq with q elements this has been solved by J. A. Green. The main goal of this project is to construct standard integral bases of these representations over integral domains in which q is invertible. The main method used is to investigate the restriction of these representations to the unitriangular subgroups U of G consisting of all lower triangular matrices having only 1 as eigenvalue. This has been successfully applied to a special class of irreducible GLn(q)-modules by Q. Guo in her thesis. The Steinberg module is another special irreducible G-module. Restricting it to U gives the regular representation of U which contains all irreducible U-modules. Using this it is hoped to confirm longstanding conjectures of Higman and Lehrer and a more recent one due to Isaacs.

有限群G的表示论中最重要的问题之一是求G的不可约复表示。对于q元有限域Fq上n阶可逆矩阵的有限一般线性群G = GLn（q），J.A.绿色。这个项目的主要目标是在q可逆的整环上构造这些表示的标准整基。所用的主要方法是调查这些表示的限制，单三角子群U的G组成的所有下三角矩阵只有1作为特征值。这已被Q成功地应用于一类特殊的不可约GLn（q）-模。郭在她的论文中。Steinberg模是另一种特殊的不可约G-模。将其限制为U给出U的正则表示，其中包含所有不可约U-模。希望利用这一点来证实希格曼和莱勒长期以来的猜想以及艾萨克斯最近提出的猜想。