On the source algebra equivalence class of blocks with cyclic defect groups, I

TL;DR

This paper explores the classification of blocks with cyclic defect groups in group algebras, showing how certain modules can be derived from character tables and identifying cases where these modules are trivial.

Contribution

It demonstrates that the endo-permutation module associated with such blocks can be determined from the character table, and identifies conditions under which this module is trivial.

Findings

Endo-permutation module can be read from the character table.

The module is trivial for certain cyclic p-blocks of specific groups.

Provides new insights into the structure of blocks with cyclic defect groups.

Abstract

We investigate the source algebra class of a p-block with cyclic defect groups of the group algebra of a finite group. By the work of Linckelmann this class is parametrized by the Brauer tree of the block together with a sign function on its vertices and an endo-permutation module of a defect group. We prove that this endo-permutation module can be read off from the character table of the group. We also prove that this module is trivial for all cyclic p-blocks of quasisimple groups with a simple quotient which is a sporadic group, an alternating group, a group of Lie type in defining characteristic, or a group of Lie type in cross-characteristic for which the prime p is large enough in a certain sense.