Coherent set of characters

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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.