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, ..., 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) + 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 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 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.

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See also

References
  • Brauer, R.; Nesbitt, C. (1941), "On the modular characters of groups", Annals of Mathematics, Second Series, 42: 556–590, doi:10.2307/1968918, 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 0460423
  • Suzuki, Michio (1955), "On finite groups with cyclic Sylow subgroups for all odd primes", American Journal of Mathematics, 77: 657–691, doi:10.2307/2372591, ISSN 0002-9327, JSTOR 2372591, MR 0074411
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