Block Structure, Fusion Systems and Conjectures of Brauer and Olsson

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Groups are a mathematical model for symmetry, both in mathematics and in areas like physics or chemistry. Representations are a tool in order to realize abstract groups in the concrete form of matrices. General representations are built up from irreducible representations, and irreducible representations of a finite group are distributed into blocks. The interplay of blocks of a finite group and blocks of proper subgroups is one of the main themes of representation theory. It is also the place of deep conjectures and thus a challenge for future research. One of the goals of this project is to determine the structure of a block, in important cases: its Morita equivalence class, the numbers of irreducible representations and their heights, Cartan invariants and decomposition numbers. The description of the block structure will often be in terms of invariants of a block like its defect group and its fusion system. A second and closely related goal of this project is to make progress on two of the central conjectures in block theory, Brauer’s Conjecture on the number of characters in a block and Olsson’s Conjecture on the number of characters of height zero.