Conjectural invariance with respect to the fusion system of an almost-source algebra

Date
2022-03-23
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Source Title
Journal of Group Theory
Print ISSN
Electronic ISSN
1435-4446
Publisher
Walter de Gruyter GmbH
Volume
25
Issue
5
Pages
973 - 995
Language
English
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Article
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Abstract

We show that, given an almost-source algebra 𝐴 of a 𝑝-block of a finite group 𝐺, then the unit group of 𝐴 contains a basis stabilized by the left and right multiplicative action of the defect group if and only if, in a sense to be made precise, certain relative multiplicities of local pointed groups are invariant with respect to the fusion system. We also show that, when 𝐺 is 𝑝-solvable, those two equivalent conditions hold for some almost-source algebra of the given 𝑝-block. One motive lies in the fact that, by a theorem of Linckelmann, if the two equivalent conditions hold for 𝐴, then any stable basis for 𝐴 is semicharacteristic for the fusion system.

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