A reduction theorem for the Alperin-McKay conjecture. (Q2842807)

The Alperin-McKay conjecture predicts that the number of irreducible characters of height 0 in a block \(B\) coincides with the number of irreducible characters of height 0 in the Brauer correspondent of \(B\).NEWLINENEWLINE The main result of the paper is a reduction theorem; it shows that the Alperin-McKay conjecture holds if every finite simple group satisfies a number of (technical) conditions. These are called the inductive AM-condition. This provides a way to prove the Alperin-McKay conjecture by making use of the classification of finite simple groups. The author shows also that various classes of finite simple groups satisfy the inductive AM-condition.NEWLINENEWLINE The paper can be seen as a refinement of earlier results by \textit{I. M. Isaacs, G. Malle} and \textit{G. Navarro} [Invent. Math. 170, No. 1, 33-101 (2007; Zbl 1138.20010)]. The author also considers the refinement of the Alperin-McKay conjecture by Isaacs and Navarro.