On blocks with trivial source simple modules

TL;DR

This paper characterizes the structure of certain blocks in modular representation theory where all simple modules of Brauer correspondents have trivial sources, extending understanding of block structures in group algebras.

Contribution

It determines the source algebra and overall structure of blocks with trivial source simple modules, building on prior work on Mathieu groups.

Findings

Identifies the source algebra for these blocks

Describes the structure of blocks without essential Brauer pairs

Provides a classification for blocks with trivial source simple modules

Abstract

Motivated by an observation in "Vertices, sources and Green correspondents of the simple modules for the large Mathieu groups", J. of Algebra 322, we determine the source algebra, and therefore all the structure, of the blocks without essential Brauer pairs where the simple modules of all the Brauer corespondents have trivial sources.