Scopes reduction and Morita equivalence classes of blocks in finite classical groups
Donovan's conjecture says that, for a prime \(l\), the \(l\)-blocks of finite groups with a fixed defect fall into finitely many Morita equivalence classes. This conjecture refines an earlier conjecture by R.~Brauer. A stronger conjecture by Puig asserts that, for any finite \(l\)-group \(D\), there are only finitely many interior \(D\)-algebras (up to isomorphism) which are source algebras of \(l\)-blocks with defect group~\(D\). \textit{J.~Scopes} has verified Donovan's conjecture for blocks of symmetric groups [J. Algebra 142, No. 2, 441-455 (1991; Zbl 0736.20008)]. \textit{L.~Puig} then showed [Algebra Colloq. 1, No. 1, 25-55 (1994; Zbl 0830.20024)] that her argument also proves his own conjecture for blocks of symmetric groups. The method was extended to other classes of groups by various authors. In the paper under review, Puig's conjecture is shown to hold for unipotent \(l\)-blocks (where \(l\) is a unitary prime) of classical groups over a fixed finite field of characteristic \(p\neq l\), with some exceptions in the case of even-dimensional orthogonal groups. Moreover, an idea is outlined which (if successful) may lead to a verification of these conjectures for all finite simple groups.
scientific article; zbMATH DE number 1512973
scientific article