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 Suzuki (1955, p. 663), based on ideas due to Brauer in (Brauer & Nesbitt 1941).

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

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 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 ij 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.