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

有限群G的表示理论中最重要的问题之一是求G的不可约复表示。对于有限域Fq上的n阶可逆矩阵的有限一般线性群G=Gln(Q)，J·A·Green已经解决了这一问题。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积分基。所用的主要方法是研究这些表示对由只有1作为特征值的所有下三角矩阵组成的G的单位角子群U的限制。这一点已成功地应用于一类特殊的不可约GLN(Q)-模，由郭启美在她的论文中提出。Steinberg模是另一种特殊的不可约G-模。把它限制在U上，给出了包含所有不可约U-模的U的正则表示。利用这一点，人们希望证实希格曼和莱勒长期以来的猜想，以及最近由艾萨克斯提出的猜想。

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.