Exceptional character

Definition

Suppose that H is a subgroup of a finite group G, and C1, ..., C**r 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, ..., C**r
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, ..., 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 φi** are given by :φi** = εs**i + a(s1 + ... + 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 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 (φi − φj)* has norm 2, so is the difference of two characters of G, which are the exceptional characters corresponding to φi and φj.

References