A bound on norms of generalized characters with applications
In the paper under review the author investigates lower bounds for \(\sum_{g\in G^\#}|\Theta(g)|^2\) where \(\Theta\) is a generalized character of \(G\). He shows that this number can not get too small unless \(\Theta\) is a multiple of the regular character. More precisely, the sum is (lower) bounded by the number of conjugacy classes of \(G\) minus 1 provided \(\Theta\) does not vanish identically on \(G^\#\). The proof depends on some kind of the Cauchy-Schwarz lemma which the author proves in a cyclotomic number field. As consequences he obtains \(k_H(G)\leq|H|\cdot|C_G(x):C_H(x)|\) for some \(x\in H^\#\) where \(H\) is a subgroup of \(G\) and \(k_H(G)\) denotes the number of irreducible characters of \(G\) not vanishing identically on \(H^\#\). If \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(k_+(G)\leq|P|\cdot|C_G(x):C_P(x)|\) for some \(x\in P^\#\) where \(k_+(G)\) denotes the number of irreducible characters of \(G\) not lying in a \(p\)-block of defect zero. Furthermore he analyzes non-trivial situations in which the sum \(\sum_{g\in G^\#}|\Theta(g)|^2\) takes minimum value.
scientific article; zbMATH DE number 1271825