Fusion category algebras.
Alperin's weight conjecture is one of the main open problems in representation theory of finite groups. Many reformulations of this conjecture exist. The paper under review presents another such reformulation. This reformulation is an equality between the number of simple modules of a \(p\)-block \(b\) of a finite group \(G\) and the number of simple modules of a certain quasi-hereditary algebra \(k\overline{\mathcal F}(b)\) over a suitable field \(k\) of characteristic \(p>0\). The construction of \(k\overline{\mathcal F}(b)\) involves the fusion system \(\mathcal F\) of \(b\), a category whose objects are the subgroups of a defect group \(P\) of \(b\). The morphisms between two objects \(Q\) and \(R\) of \(\mathcal F\) are the restrictions of certain inner automorphisms of \(G\). The construction of \(k\overline{\mathcal F}(b)\) depends also on a certain 2-cocycle \(\alpha\) on \(\mathcal F\). The author leaves the existence of \(\alpha\) open. However, Puig has pointed out that the existence of \(\alpha\) follows from unpublished work of his.
scientific article; zbMATH DE number 2102476