Autoequivalences of blocks and a conjecture of Zassenhaus
The authors deal with the Zassenhaus conjecture for certain finite simple groups. For a finite group \(G\) its integral group ring is denoted by \(\mathbb{Z} G\). The Zassenhaus conjecture claims that \(\mathbb{Z} G=\mathbb{Z} H\) as augmented algebras implies that \(a\cdot G\cdot a^{-1}=H\) for a unit \(a\in\mathbb{Q} G\). The authors verify this conjecture for \(\text{PSL}(2,p)\) with \(p\) a rational prime, for the five simple Mathieu groups, for \(A_9\) and for the sporadic Thompson group. For simple groups, the Zassenhaus conjecture is equivalent to the existence of a Zassenhaus factorization of augmented automorphisms of \(\mathbb{Z} G\): Given an augmented automorphism \(\alpha\) of \(\mathbb{Z} G\), then it has a factorization \(\alpha=\rho\cdot\gamma\), where \(\rho\in\text{Aut}(G)\) and \(\gamma\) is a central automorphism of \(\mathbb{Z} G\); i.e. it is conjugation with a unit in \(\mathbb{Q} G\) centralizing \(\mathbb{Z} G\). The proof is an interesting interplay between ordinary and modular representation theory: \(\alpha\) has a Zassenhaus factorization if and only if it acts like a group automorphism on the character table of \(G\). In order to verify this in the special examples modular representation theory is invoked: Let \(B\) be a block of \({\mathfrak k} G\) of cyclic defect (\(\mathfrak k\) must be large enough). If \(\alpha : B\to B\) is an automorphism fixing a module corresponding to an end of the Brauer-tree, then it fixes all finitely generated modules of \(B\). (Reviewer's remark: This result is definitely false if one replaces \(\mathfrak k\) by a large enough \(p\)-adic ring of integers.) This observation, together with an analysis of the character tables and decomposition matrices allows one to verify the Zassenhaus conjecture for the above mentioned groups.
scientific article; zbMATH DE number 810245
scientific article