Investigations into the Abelian Defect Group Conjecture

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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.