Brauer correspondent blocks with one simple module

TL;DR

This paper investigates the relationship between Brauer correspondent blocks in finite groups, focusing on cases where the local correspondent has a single simple module, especially for principal p-blocks with odd p.

Contribution

It characterizes the structure of principal p-blocks with a unique simple module in the context of Brauer correspondences for odd primes.

Findings

Characterization of principal p-blocks with one simple module

Insights into the structure of Brauer correspondents

Focus on odd prime p cases

Abstract

One of the main problems in representation theory is to understand the exact relationship between Brauer corresponding blocks of finite groups. The case where the local correspondent has a unique simple module seems key. We characterize this situation for the principal p-blocks where p is odd.