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Real characters in blocks. (English) Zbl 1264.20012

This paper proposes the study of “real versions” of some open problems in the modular representation theory of finite groups. These conjectures are first stated and then proved under some more special hypotheses. Real versions are presented of Brauer’s \(k(B)\)-conjecture, Olsson’s conjecture and a conjecture of Eaton. For example, the “strong real version of Brauer’s \(k(B)\)-conjecture” reads as follows: Let \(B\) be a 2-block of the finite group \(G\) with defect group \(D\). Then the number of real valued characters in \(B\) is bounded above by the number of those elements in \(D\) for which there exists an element in \(N_G(D)\) that conjugates it to its inverse. In the “weak real version” the \(N_G(D)\) is replaced by \(G\) in the above statement. The strong version is proved for symmetric groups and also when \(D\) is central, and also under some other conditions on \(D\). The last part of the paper deals with fusion of elements of defect groups.

MSC:

20C20 Modular representations and characters
20C15 Ordinary representations and characters

Software:

GAP
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Full Text: arXiv Euclid

References:

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