Investigations into the Abelian Defect Group Conjecture

有限群表示理论通过研究有限群在场上有限维向量空间上的线性作用，为对称现象提供了统一的数学模型。理解域k上有限群G的表示理论，就等于理解了群代数kG的块的表示理论，即群代数kG的不可分解直接因子。虽然特征为0的域上的理论比较好理解，但在素数特征为p的域上工作时，情况发生了巨大变化。然后，中心问题之一是G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p子群及其归一化器的表示。近年来，这一领域非常活跃，发展迅速。在他1990年具有里程碑意义的论文中，M. broou<e:1>推测，具有阿贝尔缺陷群的每一个kG块本质上与一个小得多的群代数的块具有相同的“表示理论”。本课题的目的是进一步研究这个长期存在的猜想，通过对有限群的大量序列进行验证，为这个猜想的成立提供新的证据，并改进一般证明这个猜想的方法。为了实现这一点，我们将把理论方法与计算表示理论的强大技术结合起来。

Representation theory of finite groups provides unified mathematical models for symmetry phenomena, by investigating linear actions of finite groups on finite-dimensional vector spaces over fields. 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. Whilst the theory over fields of characteristic 0 is comparatively well understood, the picture changes drastically when working over fields of prime characteristic p. One of the central questions then is to what extent the ‚global‘ representation theory of G is already controlled by ‚local‘ data, that is, representations of p-subgroups of G and their normalizers. This area has been extremely active and rich in fascinating development in recent years. In his 1990 landmark paper, M. Broué conjectured that every block of kG with an abelian defect group has essentially the ‚same‘ representation theory as a block of a much smaller group algebra. The aim of this project is to make further progress on this long-standing conjecture, by verifying it for substantial series of finite groups, to give new evidence for the conjecture to hold true, and to improve on the methods to prove the conjecture in general. To achieve this, we will combine theoretical methods with powerful techniques from computational representation theory.