arXiv 1607.087952016-07-29math.RT

具有阿贝尔亏群的块的Loewy长度

Loewy lengths of blocks with abelian defect groups

Charles W. Eaton, Michael Livesey

摘要：

本文第一部分研究了具有阿贝尔亏群的$p$ -块，证明了它的Loewy长度与指数为$p$的正规子群的块的Loewy长度之间的关系。利用这一点，我们证明了如果$B$是一个有限群的$2$ -块，它具有阿贝尔亏群$D \cong C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times (C_2)^s$，其中$a_i > 1$对所有的$i$和$r \geq 0$，那么$d < LL(B) \leq 2^{a_1}+\cdots+2^{a_r}+2s-r+1$，其中$|D|=2^d$ .当$s=1$时，上限可以改进为$2^{a_1}+\cdots+2^{a_r}+2-r$。这些一起给出了$D$的每个同构类型的尖锐上限。一个结果是，当$D$是一个交换的$2$ -群的Loewy长度是有界以上的$|D|$，除了当$D$是一个克莱因-四组和$B$是森田等价于主块的$A_5$ .我们猜想类似的界限任意素数，并给出证据，它持有的主要$3$块。

英文摘要：

We consider $p$-blocks with abelian defect groups and in the first part prove a relationship between its Loewy length and that for blocks of normal subgroups of index $p$. Using this, we show that if $B$ is a $2$-block of a finite group with abelian defect group $D \cong C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times (C_2)^s$, where $a_i > 1$ for all $i$ and $r \geq 0$, then $d < LL(B) \leq 2^{a_1}+\cdots+2^{a_r}+2s-r+1$, where $|D|=2^d$. When $s=1$ the upper bound can be improved to $2^{a_1}+\cdots+2^{a_r}+2-r$. Together these give sharp upper bounds for every isomorphism type of $D$. A consequence is that when $D$ is an abelian $2$-group the Loewy length is bounded above by $|D|$ except when $D$ is a Klein-four group and $B$ is Morita equivalent to the principal block of $A_5$. We conjecture similar bounds for arbitrary primes and give evidence that it holds for principal $3$-blocks.

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