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, ..., Cr are some conjugacy classes of H, and φ1, ..., φs are some irreducible characters of H.Suppose also that they satisfy the following conditions:

  1. s ≥ 2
  2. φi = φj outside the classes C1, ..., Cr
  3. φi vanishes on any element of H that is conjugate in G but not in H to an element of one of the classes C1, ..., Cr
  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,...,Cr is contained in H

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.

Construction

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.

See also