Block Structure, Fusion Systems and Conjectures of Brauer and Olsson

群是对称性的数学模型，在数学和物理、化学等领域都是如此。表示是一种工具，以实现矩阵的具体形式的抽象群。一般表示是由不可约表示建立的，有限群的不可约表示被分布到块中。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。这也是深层次的研究的地方，因此对未来的研究是一个挑战。该项目的目标之一是确定块的结构，在重要的情况下：它的森田等价类，不可约表示的数量及其高度，Cartan不变量和分解数。块结构的描述通常是根据块的不变量，如它的缺陷群和它的融合系统。这个项目的第二个和密切相关的目标是在块理论中的两个中心命题上取得进展，Brauer关于块中字符数的猜想和Olsson关于高度为零的字符数的猜想。

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.