Broué交换亏群猜想对Harada-Norton散在单群$HN$成立
Broué's abelian defect group conjecture holds for the Harada-Norton sporadic simple group $HN$
摘要：
在有限群的表示论中，M.布鲁埃他证明了，对于任何素数$p$，如果有限群$G$的$p$ -块$A$有一个阿贝尔亏群$P$，则$A$和它在$G$中的$P$的正规化子$N_G(P)$的Brauer对应块$B$是等价的（Rickard等价）.这个猜想被称为布鲁埃阿贝尔亏群猜想。本文证明了Broué的Abel亏群猜想对Harada-Norton单群$HN$的具有9阶初等Abel亏群$P$的非主3-块$A$成立.然后证明Broué的交换亏群猜想对所有素数$p$和Harada-Norton单群$HN$的所有$p$ -块成立。
英文摘要：
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 $P$, 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 Broué's abelian defect group conjecture. We prove in this paper that Broué'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 Broué's abelian defect group conjecture holds for all primes $p$ and for all $p$-blocks of the Harada-Norton simple group $HN$.
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