Authors:
========
Shigeo Koshitani
Department of Mathematics, Graduate School of Science, 
Chiba University, Chiba, 263-8522, Japan 

J{\"u}rgen M{\"u}ller
Lehrstuhl D f{\"u}r Mathematik, RWTH Aachen, 
52062 Aachen, Germany

Title:
======
Brou{\'e}'s abelian defect group conjecture holds
for the Harada-Norton sporadic simple group $HN$

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 $P$, then $A$ and its Brauer corresponding block $B$
of the normaliser $N_G(P)$ of $P$ in $G$ are derived equivalent
(Rickard equivalent). This conjecture is called {\it Brou{\'e}'s 
abelian defect group conjecture}. We prove in this paper that 
Brou{\'e}'s abelian defect group conjecture is true for a non-principal 
$3$-block $A$ with an elementary abelian defect group $P$ of order $9$ 
of the Harada-Norton simple group $HN$. It then turns out that 
Brou{\'e}'s abelian defect group conjecture holds for all primes $p$ 
and for all $p$-blocks of the Harada-Norton simple group $HN$.

Status:
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Submitted