Representations with a Unique Monomial Structure

Abstract In this chapter, we recall the notion of monomial structures (and their isomorphism) on a representation, show a natural monomial structure on induced representations, and introduce solitary characters (characters whose induced representation has a unique monomial structure up to isomorphism); these characters may be used to detect conjugacy of subgroups. We also recall a specific type of wreath product construction and state and prove Bart de Smit’s theorem on the existence of solitary characters for these (and a follow-up result of Pintonello for characters of degree two)—these were previously formulated and used in the context of number theory, but we present them abstractly. We give an application to covering equivalence in a very specific setup of manifolds, and also count the number of required characters, based on a formula for the commutator of a wreath product.