Character numbers and Cartan matrices of blocks with abelian defect groups

具有阿贝尔缺陷群的块的字符数和嘉当矩阵

• 批准号: 390541063
• 负责人: Privatdozent Dr. Benjamin Sambale
• 金额: --
• 依托单位: Institut für Algebra, Zahlentheorie und Diskrete Mathematik (IAZD)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/390541063?language=en
• 关键词: Character; numbers; Cartan; matrices; blocks

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.

在许多自然科学中，物体的对称性是用数学群来模拟的。在表示论中，抽象群由具体矩阵实现，以便执行计算。每一个这样的表示分解成不可约的成分，这些成分分布在块中。不可约表示本质上是由它们的特征标决定的，而这些特征标只有100个。理查德·布劳尔（Richard Brauer）指出，给定块中的字符数受局部子群的影响很大。其中有缺陷组和惯性组。这些物体之间的精确关系是Brauer、Olsson、Alperin、McKay、Dade和其他人的许多公开论文的中心话题。全球各地的地球仪研究小组正在致力于解决这些问题（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）。该项目旨在研究布劳尔的k（B）--具有阿贝尔缺陷群的块的猜想。工作时间表遵循k（GV）问题的解。此外，计划在小情况下对Broué猜想的进展。以这种方式一个积极的竞技场的有限群表示理论的重大贡献是预期的。除了布劳尔的方法和经典理论的二次形式，现代工具，如分类的有限简单的群体和新的成果互质线性群体将被应用。此外，计算机计算的建设完美的等距，同型和嘉当矩阵的目的。