Loewy lengths of centers of blocks

TL;DR

This paper refines bounds on the Loewy length of centers of blocks in finite group algebras, providing optimal bounds for abelian defect groups and improved bounds for non-abelian cases, with classifications for certain Loewy length conditions.

Contribution

It improves existing bounds on Loewy lengths of centers of blocks, especially establishing the optimal bound for abelian defect groups and enhancing bounds for non-abelian defect groups.

Findings

Optimal bound LL(ZB) ≤ LL(FD) for abelian defect groups D.

Improved bounds for non-abelian defect groups.

Classification of blocks with LL(ZB) ≥ |D|/2.

Abstract

Let B be a block of a finite group with respect to an algebraically closed field F of characteristic p>0. In a recent paper, Otokita gave an upper bound for the Loewy length LL(ZB) of the center ZB of B in terms of a defect group D of B. We refine his methods in order to prove the optimal bound LL(ZB)\le LL(FD) whenever D is abelian. We also improve Otokita's bound for non-abelian defect groups. As an application we classify the blocks B such that LL(ZB)\ge |D|/2.