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).

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:

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

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.

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.

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