The strengthened Alperin weight conjecture for \(p\)-solvable groups. (Q401749)

The well-known Alperin Weight Conjecture is an important conjecture in the area of representation theory of finite groups. We refer to the paper for a succinct description of the conjecture and related conjectures, and for what is known about it. For this review it should suffice to say that for a finite group \(G\) and a prime \(p\) the conjecture claims the existence of a bijection between the irreducible \(p\)-Brauer characters of \(G\) and the conjugacy classes of \(p\)-weights of \(G\). A recent strategy to approach this and related conjectures has been to impose various extra conditions on the bijection and thereby strengthening the conjecture in the hope that, besides of getting more information, a general proof might become easier.NEWLINENEWLINE In the paper under review the author proves a very strong form of the conjecture for \(p\)-solvable groups. This, in particular, proves, for \(p\)-solvable groups, a strengthening that had been proposed by \textit{G. Navarro} [Ann. Math. (2) 160, No. 3, 1129-1140 (2005; Zbl 1079.20010)].