Brouéʼs abelian defect group conjecture holds for the double cover of the Higman–Sims sporadic simple group

In the representation theory of finite groups, there is a well-known and important conjecture, due to Brouéʼ, saying that for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block B of the normaliser NG(P) of P in G are derived equiv...

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Bibliographic Details
Published in:Journal of algebra 2013-02, Vol.376, p.152-173
Main Authors: Koshitani, Shigeo, Müller, Jürgen, Noeske, Felix
Format: Article
Language:English
Subjects:
Abelian defect group
Brouéʼs conjecture
Double cover of the Higman–Sims sporadic simple group
Splendid derived equivalence
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Summary:In the representation theory of finite groups, there is a well-known and important conjecture, due to Brouéʼ, saying that for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block B of the normaliser NG(P) of P in G are derived equivalent. We prove in this paper, that Brouéʼs abelian defect group conjecture, and even Rickardʼs splendid equivalence conjecture are true for the faithful 3-block A with an elementary abelian defect group P of order 9 of the double cover 2.HS of the Higman–Sims sporadic simple group. It then turns out that both conjectures hold for all primes p and for all p-blocks of 2.HS.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2012.12.001