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

Abstract

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.