Loewy lengths of centers of blocks and exponents of defect groups

TL;DR

This paper introduces a new method to compute the Loewy length of the center of a block in finite group theory, providing bounds related to defect group exponents, advancing understanding of algebraic structures in modular representation theory.

Contribution

It presents a novel approach for calculating Loewy lengths of centers of blocks using subsections and defect groups, and establishes bounds based on defect group exponents.

Findings

New method for calculating Loewy length of $ZB$

Upper bounds on Loewy length depending on defect group exponents

Enhanced understanding of the structure of block centers in modular representation theory

Abstract

In this paper we study the Loewy structure of the center $ZB$ of a block of a finite group with respect to an algebraically closed field of prime characteristic. We first state a new method for calculating the Loewy length $LL(ZB)$ of $ZB$ in terms of subsections and lower defect groups. By applying this result we give an upper bound of $LL(ZB)$ which depends on the exponent of a defect group of $B$ for some cases.