Brouéʹs conjecture for non-principal 3-blocks of finite groups

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group D, then A and its Brauer correspondent p-block B of NG(D) are derived equivalent. We demonstrate in this paper that Brouéʹs conjecture holds for two non-principal 3-blocks A with elementary abelian defect group D of order 9 of the OʹNan simple group and the Higman–Sims simple group. Moreover, we determine these two non-principal block algebras over a splitting field of characteristic 3 up to Morita equivalence.