Blocks with abelian defect groups of rank 2 and one simple module

TL;DR

This paper studies blocks with abelian defect groups of rank 2 and a single simple module, establishing an isotypy with their Brauer correspondents and generalizing previous results.

Contribution

It introduces a generalization of known results by proving an isotypy between specific blocks and their Brauer correspondents.

Findings

Established an isotypy between the block and its Brauer correspondent.

Generalized previous results by Kessar and Linckelmann.

Focused on blocks with abelian defect groups of rank 2 and one simple module.

Abstract

In this paper, we investigate the block that has an abelian defect group of rank $2$ and its Brauer correspondent has only one simple module. We will get an isotypy between the block and its Brauer correspondent. It will generalize the result of Kessar and Linckelmann (\cite{KL}).