Character numbers and Cartan matrices of blocks with abelian defect groups

在许多自然科学中，物体的对称性是由数学群来模拟的。在表示论中，抽象群是通过具体的矩阵来实现的，以便进行计算。每个这样的表示被分解成不可约的成分，这些成分被分配到块中。不可约表示本质上是由它们的性质决定的，而这些性质只有有限多个。由Richard Brauer提出，给定块中的字符数量受到局部子群的强烈影响。其中包括缺陷群和惯性群。这些天体之间的精确关系是Brauer、Olsson、Alperin、McKay、Dade等人众多公开猜想的中心话题。世界各地的研究小组(美国、日本、中国、新加坡、新西兰、以色列、英国、爱尔兰、匈牙利、意大利、西班牙、法国、丹麦、瑞士、德国)都在致力于解决这些问题。这个项目旨在研究具有交换缺陷群的块的Brauer k(B)猜想。工作时间表遵循k(GV)问题的解决方案。此外，还计划在小情况下对Broué的S猜想进行改进。这样，有限群表示理论的活跃领域有望作出重大贡献。除了Brauer的方法和经典的二次型理论外，还将应用现代工具，如有限单群的分类和关于互素线性群的新结果。此外，还打算用计算机计算完全等距、等型和Cartan矩阵的构造。

In many natural sciences symmetries of objects are modeled by mathematical groups. In representation theory, abstract groups are realized by concrete matrices in order to perform computations. Every such representation decomposes into irreducible constituents which are distributed into blocks. The irreducible representations are essentially determined by their characters of which there are only finitely many. By Richard Brauer, the number of characters in a given block is strongly influenced by local subgroups. Among them are the defect group and the inertial group. The precise relationship between these objects is a central topic of numerous open conjectures by Brauer, Olsson, Alperin, McKay, Dade and others. Research groups around the globe are working on the solution of these problems (USA, Japan, China, Singapore, New Zealand, Israel, UK, Ireland, Hungary, Italy, Spain, France, Denmark, Switzerland, Germany).This project aims to investigate Brauer's k(B)-conjecture for blocks with abelian defect groups. The working schedule follows the solution of the k(GV)-problem. Moreover, progress on Broué's conjecture in small cases is planned. In this way a significant contribution to an active arena of representation theory of finite groups is expected. Apart from Brauer's methods and the classical theory of qudratic forms, modern tools like the classification of the finite simple groups and new results on coprime linear groups will be applied. Furthermore, computer calculations for the construction of perfect isometries, isotypies and Cartan matrices are intended.