Exceptional character explained

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 C₁, ..., 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:

1. s ≥ 2
2. φ_i = φ_j outside the classes C₁, ..., C_r
3. 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 C₁, ..., C_r
4. If elements of two classes are conjugate in G then they are conjugate in H
5. The centralizer in G of any element of one of the classes C₁,...,C_r is contained in H

Then G has s irreducible characters s₁,...,s_s, called exceptional characters, such that the induced characters φ_i* are given by

φ_i* = εs_i + a(s₁ + ... + s_s) + 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 s_i.

Construction

The conditions on H and C₁,...,C_r imply that induction is an isometry from generalized characters of H with support on C₁,...,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.

See also

• Dade isometry
• Coherent set of characters