On the number of characters in a block
Let \(B\) be a \(p\)-block with defect group \(D\) of a finite group \(G\), \(k_h(B)\) denotes the number of irreducible characters of height \(h\) in \(B\). The article concerns bounding the quantity \(\Delta =\sum_h p^{2h}k_h(B)\). The author proves that for \(x\in Z(D)^{\#}\) the quantity \(\Delta /|D|\) is bounded in terms of an invariant of the block \(b\) of \(C_G(x)/<x>\) with defect group \(D/<x>\). The results are somewhat technical to state, but in particular the following inequality is obtained: \(\Delta \leq |D|^2/|<x>|\). As a source of the article's subject there are two works to be pointed out of \textit{R. Brauer} [Am. J. Math. 90, 895--925 (1968; Zbl 0174.05401)] and \textit{R. Brauer} and \textit{W. Feit} [Proc. Natl. Acad. Sci. USA 45, 361--365 (1959; Zbl 0084.25801)]. [For a correction see Zbl 0743.20008.]
scientific article; zbMATH DE number 4200465