The classification of the trivial source modules in blocks with cyclic defect groups

TL;DR

This paper provides a complete classification of trivial source modules in blocks with cyclic defect groups, linking their structure to the Brauer tree and analyzing their position in the Auslander-Reiten quiver.

Contribution

It introduces a comprehensive classification of trivial source modules in blocks with cyclic defect groups, extending previous work on liftable modules.

Findings

Classification of trivial source modules in cyclic defect blocks

Description of their associated paths on the Brauer tree

Analysis of minimal distance to the boundary in the Auslander-Reiten quiver

Abstract

Relying on the classification of the indecomposable liftable modules in arbitrary blocks with non-trivial cyclic defect groups we give a complete classification of the trivial source modules lying in such blocks, describing in particular their associated path on the Brauer tree of the block in the sense of Janusz (1969). The appendix contains a description of the minimal distance from an arbitrary non-projective indecomposable liftable module to the boundary of the stable Auslander-Reiten quiver of the block.