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。虽然这个理论在特征为0的域上被很好地理解，但到目前为止，它在模情况下，即在素数特征p的域上不再成立。在这里，理解有限群G在域k上的表示理论等同于理解它的块的表示理论，即群代数kG的不可分解的直接因子。特别是，关键问题之一是G块的“全局”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-子群及其归一化器的表示。有限群的块论近年来取得了非常活跃和引人入胜的发展，但仍然充满了问题和有待解决的猜想。该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，将其作为有效的，广泛适用的工具实现，并将其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.