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

TL;DR

This paper completes the classification of source algebra equivalence classes of blocks with cyclic defect groups for quasisimple groups, focusing on invariants associated with these blocks, especially for special linear or unitary groups.

Contribution

It finalizes the classification of invariants for cyclic blocks of quasisimple groups, extending previous reductions to include special linear and unitary groups.

Findings

Classification of invariants $W( B )$ for cyclic blocks of quasisimple groups completed.

Reduced the problem to classifying blocks of special linear or unitary groups.

Established the source algebra equivalence classes for these blocks.

Abstract

This series of papers is a contribution to the program of classifying $p$-blocks of finite groups up to source algebra equivalence, starting with the case of cyclic blocks. To any $p$-block $\mathbf{B}$ of a finite group with cyclic defect group $D$, Linckelmann associated an invariant $W( \mathbf{B} )$, which is an indecomposable endo-permutation module over $D$, and which, together with the Brauer tree of~$\mathbf{B} $, essentially determines its source algebra equivalence class. In Part II of our series, assuming that $p$ is an odd prime, we reduced the classification of the invariants $W( \mathbf{B} )$ arising from cyclic $p$-blocks $\mathbf{B}$ of quasisimple classical groups to the classification for cyclic $p$-blocks of quasisimple quotients of special linear or unitary groups. This objective is achieved in the present Part III.