Block Structure, Fusion Systems and Conjectures of Brauer and Olsson

布劳尔和奥尔森的块结构、融合系统和猜想

• 批准号: 219339608
• 负责人: Professor Dr. Burkhard Külshammer
• 金额: --
• 依托单位: Lehrstuhl für Algebra und Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/219339608?language=en
• 关键词: Block; Structure; Fusion; Systems; Conjectures

Groups are a mathematical model for symmetry, both in mathematics and in areas like physics or chemistry. Representations are a tool in order to realize abstract groups in the concrete form of matrices. General representations are built up from irreducible representations, and irreducible representations of a finite group are distributed into blocks. The interplay of blocks of a finite group and blocks of proper subgroups is one of the main themes of representation theory. It is also the place of deep conjectures and thus a challenge for future research. One of the goals of this project is to determine the structure of a block, in important cases: its Morita equivalence class, the numbers of irreducible representations and their heights, Cartan invariants and decomposition numbers. The description of the block structure will often be in terms of invariants of a block like its defect group and its fusion system. A second and closely related goal of this project is to make progress on two of the central conjectures in block theory, Brauer’s Conjecture on the number of characters in a block and Olsson’s Conjecture on the number of characters of height zero.

无论是在数学领域，还是在物理或化学领域，群都是对称性的数学模型。表示法是以矩阵的具体形式实现抽象群的工具。一般表示由不可约表示建立，有限群的不可约表示被分配到块中。有限群的块和真子群的块的相互作用是表示论的主要主题之一。它也是深入猜测的地方，因此对未来的研究是一个挑战。这个项目的目标之一是确定一个块的结构，在重要的情况下：它的Morita等价类，不可约表示的数目和它们的高度，Cartan不变量和分解数。块结构的描述通常是按照块的不变量来描述的，例如它的缺陷组及其融合系统。这个项目的第二个密切相关的目标是在块论中的两个核心猜想上取得进展，这两个猜想是：布劳尔关于区块中字符的数量的猜想和奥尔森关于高度为零的字符的数量的猜想。