In mathematical finite group theory, an exceptional character of a group is a character related in a certain way to a character of a subgroup. They were introduced by, based on ideas due to Brauer in .
Suppose that H is a subgroup of a finite group G, and C1, ..., Cr are some conjugacy classes of H, and φ1, ..., φs are some irreducible characters of H.Suppose also that they satisfy the following conditions:
Then G has s irreducible characters s1,...,ss, called exceptional characters, such that the induced characters φi* are given by
φi* = εsi + a(s1 + ... + ss) + Δwhere ε is 1 or -1, a is an integer with a ≥ 0, a + ε ≥ 0, and Δ is a character of G not containing any character si.
The conditions on H and C1,...,Cr imply that induction is an isometry from generalized characters of H with support on C1,...,Cr to generalized characters of G. In particular if i≠j then (φi - φj)* has norm 2, so is the difference of two characters of G, which are the exceptional characters corresponding to φi and φj.