Central ideals and Jacobson radicals of blocks of group algebras

TL;DR

This paper investigates the relationship between central ideals and Jacobson radicals in blocks of group algebras, characterizing blocks where certain powers of the radical are non-zero central ideals.

Contribution

It provides a characterization of blocks with the property that the square or cube of the radical is a non-zero central ideal.

Findings

Characterization of blocks with radical square as a non-zero central ideal

Analysis of blocks with radical cube as a non-zero central ideal

Insights into the structure of group algebra blocks

Abstract

We study the relationship between the central ideals and the Jacobson radicals of blocks of group algebras.In particular, we characterize the blocks with the property that the square of the radical is a non-zero central ideal. Moreover, we also consider the blocks with the property that the cube of the radical becomes a non-zero central ideal.