Loewy lengths of blocks with abelian defect groups

TL;DR

This paper investigates the Loewy lengths of blocks with abelian defect groups, establishing bounds based on group structure and providing sharp bounds for various cases, including a conjecture for primes beyond 2.

Contribution

It introduces new bounds for Loewy lengths of blocks with abelian defect groups and explores their sharpness and implications, including a conjecture for prime cases beyond 2.

Findings

Established bounds for 2-blocks with abelian defect groups.

Identified cases where bounds are sharp, including Klein-four groups.

Conjectured similar bounds for principal blocks at prime 3.

Abstract

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.