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On saturated fusion systems and Brauer indecomposability of Scott modules

Kessar, R., Kunugi, N. and Mitsuhashi, N. (2011). On saturated fusion systems and Brauer indecomposability of Scott modules. Journal of Algebra, 340(1), pp. 90-103. doi: 10.1016/j.jalgebra.2011.04.029


Let $p$ be a prime number, $G$ a finite group, $P$ a $p$-subgroup of $G$ and $k$ an algebraically closed field of characteristic $p$. We study the relationship between the category $\Ff_P(G)$ and the behavior of $p$-permutation $kG$-modules with vertex $P$ under the Brauer construction. We give a sufficient condition for $\Ff_P(G)$ to be a saturated fusion system. We prove that for Scott modules with abelian vertex, our condition is also necessary. In order to obtain our results, we prove a criterion for the categories arising from the data of $(b, G)$-Brauer pairs in the sense of Alperin-Brou\'e and Brou\'e-Puig to be saturated fusion systems on the underlying $p$-group.

Publication Type: Article
Subjects: Q Science > QA Mathematics
Departments: School of Mathematics, Computer Science & Engineering > Mathematics
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Date available in CRO: 23 Nov 2012 12:55
Date of first online publication: September 2011
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