有限群代数的几何结构与自对偶模

In our last finished project "Determinations and Applications of Hyperbolic Modules for Group Algebras", the proposal and Prof. Fan Yun completely solved the criteria of hyperbolic modules over a finite field of characteristic 2. The result was published on "Advances in Mathematics" and the reviewer commented that "The main results are quite surprising and nice with interesting applications in representation and coding theory. They definitely should go in print". The present project is a follow-up work of determinations of hyperbolic modules. We will study the anisotropic modules, Witt groups of group algebras, and Grothendieck groups of self-dual modules. In particular, we will describe the properties and behaviors of the above two kind of groups under the extension of ground fields and inductions of subgroups, respectively. The results obtained will be used in the study of related problems in representation theory of groups and in algebraic coding.

在结题项目 “群代数的双曲模判别及应用” 中，申请人和樊恽合作完整地解决了特征 2 域上双曲模的判别问题，所得结果于 2015 年发表在 Advances in Mathematics 上，审稿人评论说 “The main results are quite surprising and nice with interesting applications in representation and coding theory. They definitely should go in print”。 本项目属双曲模判别问题的后续研究课题，拟研究反迷向模的判别问题、群代数的 Witt 群以及自对偶模的 Grothendieck 群，特别是描述这两个群分别在基域扩张和子群诱导下的性质和表现，所得结果将用来研究群表示论和代数编码中若干相关的重要问题。

本项目主要研究有限群表示论中的几何结构与自对偶模，特别是群代数的Witt群以及自对偶模的Grothendieck群及其在基域扩张和子群诱导下的性质和表现，属于很新的前沿课题。但Witt群的研究可归结为Witt核平凡的辛模，亦即反迷向模的研究；而自对偶模的Grothendieck群的研究本质上是考察自对偶模的诱导和限制等性质。..具体来讲，本项目深入探讨了辛模的反迷向判据及其在本原特征标问题中的若干应用、双曲模的性质及其在单项特征标和M-群诸多重要问题中的应用、自对偶模与其Grothendieck群的表现以及在扩张问题和正则轨道中的应用。这三个问题都是有限群表示论中的重要研究课题，本项目取得的成果具有重要的理论意义和应用价值，并为研究相关的表示论问题提供了崭新的观察视角和证明技术。