Coherent pair

Definition
Let $$N$$ be a normal subgroup of a group $$G$$. Let $$D$$ be a set of characters of $$N$$, where each character is a positive linear combination of irreducible characters. Denote by $$\Z_0(S)$$ the set of those integral linear combinations $$\alpha$$ of members of $$S$$, such that $$\alpha(1) = 0$$.

Suppose there exists a linear isometry:

$$\tau: \Z_0(S) \to Ch(G)$$

where $$Ch(G)$$ is the character ring of $$G$$. Then we say that $$(S,\tau)$$ is a coherent pair.

We call $$S$$ a coherent set of characters of there exists a $$\tau$$ such that $$(S,\tau)$$ is a coherent pair.

$$\tau$$ is often induction from $$N$$ to $$G$$, or something closely related to induction.