On the source algebra equivalence class of blocks with cyclic defect groups, II
Gerhard Hiss, Caroline Lassueur
TL;DR
The paper advances the program of classifying p-blocks up to source-algebra equivalence for cyclic defect groups by focusing on the invariant W(B) and its realization as an indecomposable endo-permutation module. It develops a comprehensive framework of preliminaries—Dade groups, Brauer correspondences, and reductive-group theory—and proves reduction theorems that translate the problem for quasisimple classical groups to the general linear and unitary groups. In GL and GU, the authors provide explicit determinations of W(B): it is typically the trivial module, with a precise nontrivial exception in the unitary-like case when p divides q+1 and n is odd with p≡−1 (mod 4), where W(B) is described in terms of p, q, n. Notably, the work yields concrete examples where Morita equivalences do not preserve source algebras, illustrating the subtleties in the source-algebra classification and setting the stage for Parts III and IV to complete the classification for SL/PSL and exceptional 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.