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 $\mathbb {Z} _{0}(S)$ the set of those integral linear combinations $\alpha$ of members of $S$, such that $\alpha (1)=0$.

Suppose there exists a linear isometry:

$\tau :\mathbb {Z} _{0}(S)\to Ch(G)$

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.