Brou\'e's conjecture for non-principal 3-blocks of finite groups

Shigeo Koshitani, Naoko Kunugi, and Katsushi Waki

Preprint, April 2001

Abstract. In representation theory of finite groups, there is a well known and
important conjecture due to M. Brou\'e. 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 $N_G(D)$ are derived 
equivalent. We demonstrate in this paper that the Brou\'e's conjecture holds for
two non-principal 3-blocks $A$ with elementary abelian defect group $D$ of order
9 of the O'Nan 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.