有限群代数的几何结构与自对偶模

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。

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.