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Trivial source characters in blocks with cyclic defect groups. (English) Zbl 1462.20004

Let \(G\) be a finite group and \(p\) be a prime number such that \(p | |G|\), and let \(k\) be an algebraically closed field of characteristic \(p \geq 3\). Moreover, assume that we are given a \(p\)-modular system \((K, \mathcal{O}, k)\) which is large enough for \(G\) and all of its subgroups and quotients. The main result of this paper is to provide a complete description of the ordinary characters of the trivial source modules lying in a \(p\)-block \(B\) with a non-trivial cyclic defect group \(D\). (The actual statement of the main result is too long and technical to be stated here). The proof uses the recent classification of these trivial source modules in terms of paths on the Brauer tree by G. Hiss and the second author [“The classification of the trivial source modules in blocks with cyclic defect groups”, Preprint]. In particular, it is shown how to recover the exceptional constituents of such characters using the source algebra of the block. The main result generalizes earlier results by the first author and N. Kunugi [Math. Z. 265, No. 1, 161–172 (2010; Zbl 1211.20014)] as well as M. Takahashi [J. Algebra 353, No. 1, 298–318 (2012; Zbl 1252.20007)].

MSC:

20C20 Modular representations and characters

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References:

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