有限一般线性群的unipotent Specht模与Supercharacter理论

表示理论是当前国际数学研究的前沿课题，其中有限群的模表示理论对有限单群的分类问题起了至关重要的作用。该分类问题解决后，有限群模表示论的主要问题常归结为有限李型单群的表示问题。有限一般线性群G是有限李型群中的一类，考虑到它的许多表示性质与对称群的表示性质类似，Dipper-James提出了G的Unipotent Specht模的标准基猜想。本项目将围绕这一猜想展开研究。在前期研究中本人已证明该猜想在某些特殊情况下是正确的。本项目拟将该证明方法进行推广，预计通过对Unipotent Specht模在幂幺下三角群U上的限制的研究，结合近年来由André-Yan引入的supercharacter理论，构造出与对称群的标准基类似的Unipotent Specht模的标准基。另外，本项目还将研究U及其子群的supercharacter理论，以期对Higman关于U的共轭类个数的猜想起到推动作用。

The representation theory is one of the most attractive subjects in the international mathematical research fields. In particular, the representation theory of the finite groups played an important role in the classification of the finite simple groups. After solving this difficulty, the main problems in the representation theory of the finite groups can be reduced to the ones in the representation theory of the finite simple Lie groups. The finite general linear group G is one type of the finite Lie groups. Since some properties of the representation theory of G is quite similar as the ones of the representation theory of the symmetric groups, Dipper-James put forward a conjecture concerning the standard bases of the unipotent Specht modules for the finite general linear groups. In this project, we shall focus on this conjecture. In the previous work of the applicant, this conjecture was proved for some special cases. Hence it is natural to consider a generalization of the method there to the general cases. Through investigating the restriction of the unipotent Specht modules to the lower unitriangular groups U and the U-supercharacter theory introduced by André-Yan, the standard bases of the unipotent Specht modules are hoped to be constructed. Moreover, the supercharacter theory of U and its subgroups will also be studied and we believe this will shed some light on the longstanding conjecture of Higman, which concerned on the number of the conjugacy classes of U.