整群环的单位刻划

A dichotomy for integral group rings via higher modular groups as amalgamated products

We show that $\mathcal{U}(\mathbb{Z}G)$, the unit group of the integral group ring $\mathbb{Z} G$, either satisfies Kazhdan's property (T) or is, up to commensurability, a non-trivial amalgamated product, in case $G$ is a finite group satisfying some mild conditions. Crucial in the proof is the construction of amalgamated decompositions of the elementary group $\operatorname{E}_2(\mathcal{O})$, where $\mathcal{O}$ is an order in a rational division algebra. A major step is to introduce subgroups $\operatorname{E}_2(Γ_n(\mathbb{Z}))$ inside the so-called higher modular groups $\operatorname{SL}…

阿贝尔数域上群代数的例外分支的分类

A classification of exceptional components in group algebras over abelian number fields

当考虑$\mathcal{O}_F G$的单位群（$\mathcal{O}_F$是阿贝尔数域$F$和有限群$G$的整数环）时，$FG$的Wedderburn分解中的某些成分会对已知的单位的一般构造造成问题;这些成分被称为例外。例外环分为两类：第一类是除环，第二类是$2 \times 2$ -矩阵环。对于类型1的例外分支，我们通过描述极小群的七种情况（w.r.t.在$F$上的Wedderburn分解中具有那些除环的类）。通过描述$FG$中所有58个具有忠实例外Wedderburn分支的有限群$G$，对数域$F$上有限群的群代数中出现的2型例外分支进行了分类.

作用于张量积的群

Groups acting on tensor products

保持分配积的群在代数中经常遇到。例子包括结合环和非结合环的自同构群、典型群和p-群的自同构群。虽然这样的乘积的多样性排除了描述保持它们的群的一般结构的任何现实的希望，但有理由期望，可以从对普遍分配乘积的考察中获得洞察力：张量积。本文给出了半单环和半准素环上保张量积群的详细描述，并给出了构造这些群的生成元的有效算法。我们还讨论了我们的方法的算法问题，所有目前已知的方法需要一个指数的工作量的应用。

Group rings and character sums: tricks of the trade

The combination of the group ring setting with the methods of character theory allows an elegant and powerful analysis of various combinatorial structures, via their character sums. These combinatorial structures include difference sets, relative difference sets, partial difference sets, bent functions, hyperplanes, spreads, and LP-packings. However, the literature on these techniques often relies on peculiar conventions and implicit understandings that are not always readily accessible to those new to the subject. While there are many excellent advanced sources describing these techniques, we…