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

TL;DR

This paper advances the classification of blocks with cyclic defect groups in finite groups by analyzing endo-permutation modules and reducing the problem for classical groups to general linear and unitary groups.

Contribution

It reduces the classification problem for classical groups of Lie type to that for general linear and unitary groups, completing the classification for the latter.

Findings

Classification reduced to general linear and unitary groups

Complete classification achieved for general linear and unitary groups

Progress in classifying blocks with cyclic defect groups

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 Parts II-IV of our series of papers, we classify, for odd $p$, those endo-permutation modules of cyclic $p$-groups arising from $p$-blocks of quasisimple groups. In the present Part II, we reduce the desired classification for the quasisimple classical groups of Lie type $B$, $C$, and $D$ to the corresponding objective for the general linear and unitary groups; the classification is completed for the latter groups.