Exceptional character
# 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).

## Definition

Suppose that H is a subgroup of a finite group G, and $C_1,$ ..., $C_r$ are some conjugacy classes of H, and $φ_1,$ ..., $φ_s$ are some irreducible characters of H. Suppose also that they satisfy the following conditions:

* s ≥ 2
* $φ_i$ = $φ_j$ outside the classes $C_1,$ ..., $C_r$
* $φ_i$ vanishes on any element of H that is conjugate in G but not in H to an element of one of the classes $C_1,$ ..., $C_r$
* If elements of two classes are conjugate in G then they are conjugate in H
* The centralizer in G of any element of one of the classes $C_1,...,C_r$ is contained in H

Then G has s irreducible characters $s_1,...,s_s,$ called exceptional characters, such that the induced characters $φ_i*$ are given by

    $φ_i*$ = $εs_i$ + $a(s_1$ + ... + $s_s)$ + Δ

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 $φ_j.$

