Generalized Clifford correspondences for group characters
Suppose that \(\phi\) is a complex irreducible character of a normal subgroup N of a finite group G. Clifford Theory tells us that induction is a bijection of the set \(Irr(G\{\phi \}| \phi)\) (of all irreducible characters \(\chi\) of the stabilizer \(G\{ \phi\}\) of \(\phi\) in G such that \(\chi\) lies over \(\phi)\) onto the corresponding set \(Irr(G| \phi)\) for G. In this paper we consider the consequences of the existence of such a bijection for an arbitrary irreducible character of an arbitrary subgroup of G. More specifically, we define an inductive source for G to be any irreducible character \(\phi\) of a subgroup of G such that induction is a bijection of \(Irr(G\{\phi \}| \phi)\) onto \(Irr(G| \phi),\) and the associated IS-correspondent \(\psi\{ \phi\}\) of any \(\psi \in Irr(G| \phi)\) to be the unique character in \(Irr(G\{\phi \}| \phi)\) from which \(\psi\) is induced. We show that these sources and correspondents have most of the usual properties of irreducible characters of normal subgroups and the associated Clifford correspondents, including the correct ''compounding properties''. Indeed the IS-correspondents are much easier to handle then (repeated) Clifford correspondents due to their closure under most constructions. We also show that IS-correspondents reduce to repeated Clifford correspondents in certain interesting cases of nilpotent and odd normal subgroups.
irreducible characters of normal subgroups
scientific article; zbMATH DE number 3849466