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 , 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. 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 irreducible characters of H. Write I(S) for the set of integral linear combinations of S, and I₀(S) for the subset of degree 0 elements of I(S). Suppose that τ is an isometry from I₀(S) to the degree 0 virtual characters of G. Then τ is called coherent if it can be extended to an isometry from I(S) to characters of G and I₀(S) is non-zero. Although strictly speaking coherence is really a property of the isometry τ, it is common to say that the set S is coherent instead of saying that τ is coherent.

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 normalizer N, such that N is a Frobenius group with kernel H, and let S be the irreducible characters of N that do not have H in their kernel. Suppose that τ is a linear isometry from I₀(S) into the degree 0 characters of G. Then τ is coherent unless

• 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(F_2ⁿ) for n>1 and H is a Sylow 2-subgroup, with τ induction, then coherence fails for the first reason: H is elementary abelian and N/H has order 2ⁿ–1 and acts simply transitively on it.

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.

References