Computational aspects of block theory of finite groups

有限群表示理论通过研究有限维向量空间上的线性群作用，为对称现象提供了统一的数学模型。虽然这一理论在特征为0的域上已被很好地理解，但这在模的情况下，即在素特征p的域上不再成立。在这里，理解域k上的有限群G的表示理论等同于理解它的块的表示理论，即群代数Kg的不可分解的直因子。具体而言，一个关键问题是G块的“整体”表示理论在多大程度上已经由“局部”数据控制，即G的非平凡p-子群及其正规化子的表示。近年来，有限群的块论非常活跃，并取得了许多引人入胜的发展，但仍然充满了疑问和开放猜想。这个项目的目的是通过开发计算技术来处理现代有限群块理论中突出的代数对象，将它们作为有效的、广泛应用的工具来实现，并将它们应用于大量有趣的例子，从而为这一领域做出贡献。

Representation theory of finite groups provides unified mathematical models for symmetry phenomena, by investigating linear group actions on finite-dimensional vector spaces. While the theory is fairly well-understood over fields of characteristic 0, this by far does no longer hold in the modular case, that is, over fields of prime characteristic p. Here, understanding the representation theory of a finite group G over a field k is equivalent to understanding the representation theory of its blocks, that is, the indecomposable direct factors of the group algebra kG. In particular, one of the key questions is to what extent the `global' representation theory of a block of G is already controlled by ‘local’ data, that is, representations of non-trivial p-subgroups of G and their normalizers. Block theory of finite groups has been extremely active and rich in fascinating developments in recent years, but still is full of questions and open conjectures. The aim of this project is to contribute to this area, by developing computational techniques to handle the algebraic objects featuring prominently in modern block theory of finite groups, implementing them as efficient, widely applicable tools, and applying them to substantial interesting examples.