Character numbers and Cartan matrices of blocks with abelian defect groups

点击翻译按钮获取中文摘要

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.