Coherent set of characters

Definition

Let $N$ be a normal subgroup of a finite group $G$. Let $S$ be a set of characters of $N$ (each character being a positive linear combination of irreducible characters). Let ${\mathbb{Z}}_0(S)$ denote the set of those integral linear combinations $\alpha$ of elements of $S$ such that $\alpha (1)=0$.

We call $S$ a coherent set of characters if there exists a linear isometry $\tau$ from ${\mathbb{Z}}_0(S)$ to the character ring of $G$. Also $(S,\tau )$ is termed a coherent pair.