Counting characters in blocks. I
The author presents a formula expressing the number of ordinary irreducible characters of a given height in a \(p\)-block \(B\) of a finite group \(G\) in terms of the numbers of ordinary irreducible characters of related heights in related blocks of certain \(p\)-local subgroups of \(G\). He shows by examples that this formula can fail if \(O_ p(G)\neq 1\) or if \(B\) has defect zero. Nevertheless, he conjectures that it holds in all other cases. Using results of \textit{R. Knörr} and \textit{G. Robinson} [J. Lond. Math. Soc., II. Ser. 39, 48-60 (1989; Zbl 0672.20005)], he shows that this conjecture would imply the Alperin weight conjecture. He verifies the conjecture for blocks with cyclic defect groups, and for all blocks of either the Mathieu group \(M_{11}\) or the Janko group \(J_ 1\). These easy verifications may actually be part of the eventual proof of a stronger version of the conjecture. The author claims that one such strong conjecture holds in general if it holds for each finite simple group \(G\). He also says that that strong conjecture coincides with the present one when \(G\) has trivial outer automorphism group and trivial Schur multiplier. Both \(M_{11}\) and \(J_ 1\) are among the eleven sporadic simple groups satisfying these two conditions. He hopes that someone will test the conjecture for the other nine, probably with computers for the largest ones.
numbers of ordinary irreducible characters
scientific article; zbMATH DE number 4216119
scientific article