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Quasi-isolated blocks and Brauer’s height zero conjecture. (English) Zbl 1317.20006

This paper has two main results. Firstly, the authors complete the parametrisation of all \(p\)-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups. In other cases, such parametrisation was provided before in works of many investigators.
The second main result is the proof of one direction (the ‘if part’) of Brauer’s following long-standing height zero conjecture (1955): A \(p\)-block \(B\) of a finite group has an abelian defect group if and only if every ordinary irreducible character in \(B\) has height zero. The proof uses the reduction of this direction by T. R. Berger and R. Knörr [Nagoya Math. J. 109, 109-116 (1988; Zbl 0637.20006)] to the case of quasi-simple groups and also the first main result of the paper. The authors also apply their results on blocks to verify a conjecture of G. Malle and G. Navarro on nilpotent blocks [see Trans. Am. Math. Soc. 363, No. 12, 6647-6669 (2011; Zbl 1277.20013)] for all quasi-simple groups.

MSC:

20C20 Modular representations and characters
20C15 Ordinary representations and characters
20C33 Representations of finite groups of Lie type

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