Vertices for characters of \(p\)-solvable groups (Q2782662)

Let \(p\) be a prime, and let \(G\) be a finite \(p\)-solvable group. The author associates to every \(\chi\in\text{Irr}(G)\) a pair \((Q,\delta)\) where \(Q\) is a \(p\)-subgroup of \(G\) and \(\delta\in\text{Irr}(Q)\). The author calls \((Q,\delta)\) a vertex of \(\chi\). He shows that \(\chi\) and \(\delta\) have the same \(p\)-defect, that \(Q=O_p(N_G(Q,\delta))\) and that \(\chi(g)=0\) whenever the \(p\)-part of \(g\) is not contained in a conjugate of \(Q\). Let \(P_0:=Q\) and \(P_{i+1}:=O_p(N_G(P_i))\) for \(i>0\). The final term \(R:=P_n\) of the chain \(Q=P_0\trianglelefteq P_1\trianglelefteq\cdots\trianglelefteq P_n=P_{n+1}=\cdots\) is called the radical closure of \(Q\) in \(G\). The author shows that the induced character \(\delta^R\) is irreducible, and that there is a defect group \(D\) of the \(p\)-block containing \(\chi\) such that \(Q\subseteq R\subseteq D\) and \(C_D(Q)=Z(Q)\).NEWLINENEWLINENEWLINERestriction to \(p\)-regular elements defines a bijection between all \(\chi\in\text{Irr}(G)\) with vertex \((Q,1_Q)\) and all irreducible \(p\)-modular characters of \(G\) with vertex \(Q\) (in Green's sense). If \(Q\) is normal in \(G\) and \(\delta\) is \(G\)-invariant then \(\chi\in\text{Irr}(G)\) has vertex \((Q,\delta)\) if and only if \(\delta\) is a constituent of the restiction \(\chi_Q\) and \(\chi(1)_p=\delta(1)_p|G:Q|_p\). The author explains how these results are related to the Fong-Swan theorem, to Alperin's weight conjecture, to the McKay conjecture and to a conjecture of Robinson.