Morita equivalence for blocks of finite general linear groups
In modular representation theory of finite groups, one of the prominent problems is to find invariants of a block of a group algebra which serve as a measure for the complexity of its module category. Consequently, one may inquire how well these invariants control the structure of the module category of a block. One such invariant is the defect group of a block. P. Donovan [\textit{J. P. Alperin}, Proc. Symp. Pure Math. 37, 369-375 (1980; Zbl 0449.20019)]\ conjectured that up to Morita equivalence there is only a finite number of block algebras with defect group isomorphic to a given \(r\)-group. The conjecture was answered affirmatively by J. Scopes in 1990 for blocks of the symmetric groups by using combinatorial techniques of the representation theory of these groups. In the paper under review, by generalizing Scopes' method, the author shows that for a given weight vector \(w\) and a given \(q\) there is only a finite number of Morita equivalence classes of blocks with weight vector \(w\), for \(R\text{GL}(n,q)\) in the nondescribing characteristic. He considers \(R\in\{{\mathfrak R},F\}\), where \({\mathfrak R}\) is a complete discrete valuation ring whose residue class field \(F\) is an algebraically closed field whose characteristic \(r\) is coprime to \(q\).
scientific article; zbMATH DE number 980703
scientific article