Computational aspects of block theory of finite groups

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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.