Coherent set of characters
In mathematical representation theory, coherence is a property of sets of characters that allows one to extend an isometry from the degree-zero subspace of a space of characters to the whole space. The general notion of coherence was developed by Feit (1960, 1962), as a generalization of the proof by Frobenius of the existence of a Frobenius kernel of a Frobenius group and of the work of Brauer and Suzuki on exceptional characters. Feit & Thompson (1963, Chapter 3) developed coherence further in the proof of the Feit–Thompson theorem that all groups of odd order are solvable.
Definition
Suppose that H is a subgroup of a finite group G, and S a set of
Feit's theorem
Feit proved several theorems giving conditions under which a set of characters is coherent. A typical one is as follows. Suppose that H is a subgroup of a group G with
- either H is an elementary abelian group and N/H acts simply transitively on its non-identity elements (in which case I0(S) is zero)
- or H is a non-abelian p-group for some prime p whose abelianization has order at most 4|N/H|2+1.
Examples
If G is the simple group SL2(F2n) for n>1 and H is a Sylow 2-subgroup, with τ induction, then coherence fails for the first reason: H is
If G is the simple Suzuki group of order (2n–1) 22n( 22n+1) with n odd and n>1 and H is the Sylow 2-subgroup and τ is induction, then coherence fails for the second reason. The abelianization of H has order 2n, while the group N/H has order 2n–1.
Examples
In the proof of the Frobenius theory about the existence of a kernel of a Frobenius group G where the subgroup H is the subgroup fixing a point and S is the set of all irreducible characters of H, the isometry τ on I0(S) is just induction, although its extension to I(S) is not induction.
Similarly in the theory of exceptional characters the isometry τ is again induction.
In more complicated cases the isometry τ is no longer induction. For example, in the Feit–Thompson theorem the isometry τ is the Dade isometry.