authors:
Shigeo Koshitani, J\"urgen M\"uller, Felix Noeske

title:
Brou{\'e}'s 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 Brou{\'e}', 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 $N_G(P)$ of $P$ in $G$ are derived equivalent.
We prove in this paper, that Brou{\'e}'s abelian defect group 
conjecture, and even Rickard's 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.{\sf 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.{\sf HS}.

status:
submitted