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 $$\Z_0(S)$$ 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 $$\Z_0(S)$$ to the character ring of $$G$$. Also $$(S,\tau)$$ is termed a coherent pair.