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Some problems in computational representation theory. (English) Zbl 0748.20005

Several problems are presented which occur in the Computational Representation Theory. The main ideas are shown how to solve the stated problems by explicit computations, and references to (partial) solutions or specific examples are given, where these problems have been solved.
First, the author deals with the computation of character tables of finite groups of Lie type (especially \(E_ n(q)\) for \(n=7,8\)) using Deligne-Lusztig characters and Green functions. Then, methods are reviewed for the construction of permutation modules by computing complete double coset representative systems and for the extension of representations of pairs of proper subgroups. Now the decomposition of modules (or – eventually – the proof of their indecomposability) in the modular case is discussed. This section includes a discussion of methods to calculate the socle series, vertices, and sources of modules. Also some remarks about the induction and restriction of modular representations are made.
Finally the Brauer question is mentioned whether the number of characters of a \(p\)-block \(B\) is bounded by the order of its defect group. As a demonstration of the power of the previously listed algorithms and as a special example to support Brauer’s assumption the socle series of the Green correspondents of the simple modules of the simple Tits group \(^ 2F_ 4(2)'\) are given at the end of the article.
Reviewer: M.Weller

MSC:

20C20 Modular representations and characters
68W30 Symbolic computation and algebraic computation
20C40 Computational methods (representations of groups) (MSC2010)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C33 Representations of finite groups of Lie type
20C34 Representations of sporadic groups
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