McKay natural correspondences on characters.
Let \(G\) be a finite group, \(p\) a prime, \(P\in\text{Syl}_p(G)\) and \(N=N_G(P)\). The well-known McKay conjecture states that if \(\text{Irr}_{p'}(G)\) is the set of irreducible (complex) characters of \(p'\)-degree of \(G\), then \(|\text{Irr}_{p'}(G)|=|\text{Irr}_{p'}(N)|\). It is well known that, in general, no choice-free correspondence can exist between \(\text{Irr}_{p'}(G)\) and \(\text{Irr}_{p'}(N)\). In the paper under review the authors study situations, where a natural correspondence exists. The main result is the following Theorem. Let \(p\) be odd. Suppose that \(N=PC_G(P)\). If \(\chi\in\text{Irr}_{p'}(G)\) lies in the principal block \(B_0(G)\) of \(G\), then \(\chi_N=\chi^*+\Delta\), where \(\chi^*\in\text{Irr}(N)\) is linear in the principal block and \(\Delta\) is either zero or a character whose irreducible constituents all have degree divisible by \(p\). Furthermore, the map \(\chi\mapsto\chi^*\) is a bijection \(\text{Irr}_{p'}(B_0(G))\to\text{Irr}_{p'}(B_0(N))\) where \(\text{Irr}_{p'}(B_0(G))\) is the set of irreducible characters in \(B_0(G)\) of degree not divisible by \(p\). The authors point out that the Theorem is not true for \(p=2\) (as \(S_5\) shows); moreover, it is not true for \(p\)-blocks of maximal defect. The authors discuss several consequences of the theorem and also show that for \(p\)-solvable groups, there even exists a natural correspondence between \(\text{Irr}_{p'}(G)\) and \(\text{Irr}_{p'}(N)\) which extends both the correspondence of the Theorem above and the Glauberman correspondence.
scientific article; zbMATH DE number 6382482
scientific article