Investigations into the Abelian Defect Group Conjecture

有限群表示论通过研究域上有限维向量空间上有限群的线性作用，为对称现象提供了统一的数学模型。理解域k上的有限群G的表示论等价于理解其块的表示论，即群代数kG的不可分解的直接因子。虽然理论上的领域的特点0是比较好的理解，图片急剧变化时，工作领域的素数特征p.一个中心问题，然后是在何种程度上的"全球“表示理论的G已经控制的”本地“数据，即表示的p-子群的G和他们的正规化。近年来，这一领域一直非常活跃，富有迷人的发展。在他1990年具有里程碑意义的论文中，M. Broué指出，kG的每一个具有阿贝尔亏群的块，本质上都具有与一个小得多的群代数的块“相同”的表示理论。该项目的目的是进一步推进这一长期存在的猜想，通过验证它的大量系列的有限群，给出新的证据，猜想成立，并改进的方法来证明猜想一般。为了实现这一点，我们将结合联合收割机的理论方法与强大的技术，从计算表示理论。

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.