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A note on module structures of source algebras of block ideals of finite groups. (English) Zbl 1477.20023

Summary: Let \(b\) be a block ideal of the group algebra of a finite group \(G\) over an algebraically closed field \(k\) of prime characteristic \(p\) with a defect group \(P\). Some direct summands, as \(k[P\times P]\)-module, of a source algebra of the block ideal \(b\) outside of the inertial group of a maximal \(b\)-Brauer pair will be given; their multiplicities are congruent to 1 modulo \(p\).

MSC:

20C20 Modular representations and characters
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References:

[1] Külshammer, B., Lectures on Block Theory, London Math. Soc. Lecture Note Ser., vol. 161 (1991), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0726.20006
[2] Puig, L., The hyperfocal subalgebra of a block, Invent. Math., 141, 365-397 (2000) · Zbl 0957.20007
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