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 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) + 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 (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

References

Brauer, R.; Nesbitt, C. (1941), "On the modular characters of groups", Annals of Mathematics. Second Series 42: 556–590, ISSN 0003-486X, JSTOR 1968918, MR 0004042
Isaacs, I. Martin (1994), Character Theory of Finite Groups, New York: Dover Publications, ISBN 978-0-486-68014-9, MR 460423
Suzuki, Michio (1955), "On finite groups with cyclic Sylow subgroups for all odd primes", American Journal of Mathematics 77: 657–691, ISSN 0002-9327, JSTOR 2372591, MR 0074411

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