On the connected components of the spectrums of the character rings of a finite group (Q536185)

Let \(G\) be a finite group and \(K\) a subfield of the field \(\mathbb C\) of complex numbers. Denote by \(R(G)\) the ring of \(\mathbb Z\)-linear combinations of irreducible complex characters of \(G\), by \(R_K(G)\) the ring of \(\mathbb Z\)-linear combinations of irreducible \(K\)-characters of \(G\), and by \(\bar R_K(G)\) the ring of functions in \(R(G)\) which take values in \(K\). In this paper, the authors determine the number of connected components of the spectrums of three kinds of character rings of the forms \(S\otimes_{\mathbb Z}R_K(G)\) and \(S\otimes_{\mathbb Z}\bar R_K(G)\) in terms of the numbers of certain of \(\pi\)-regular conjugacy classes of \(G\), where \(\pi\) is a set of rational primes.