Exceptional character

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 .
Definition.
Suppose that "H" is a subgroup of a finite group "G", and "C"1, ..., "C""r" are some conjugacy classes of "H", and 
φ1, ..., φ"s" are some irreducible characters of "H".
Suppose also that they satisfy the following conditions:
Then "G" has "s" irreducible characters "s"1...,"s""s", called exceptional characters, such that the induced characters φ"i"* are given by
where 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 "s""i".
Construction.
The conditions on "H" and "C"1...,"C""r" imply that induction is an isometry from generalized characters of "H" with support on "C"1...,"C""r" to generalized characters of "G". In particular if "i"≠"j" then (φ"i" − φ"j")* has norm 2, so is the difference of two characters of "G", which are the exceptional characters corresponding to φ"i" and φ"j".