# 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.

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 *I*_{0}(*S*) for the subset of degree 0 elements of *I*(*S*). Suppose that τ is an isometry from *I*_{0}(*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 *I*_{0}(*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 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 *I*_{0}(*S*) into the degree 0 characters of *G*. Then τ is coherent unless

If *G* is the simple group SL_{2}(**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 2^{n}–1 and acts simply transitively on it.

If *G* is the simple Suzuki group of order (2^{n}–1) 2^{2n}( 2^{2n}+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 2^{n}, while the group *N*/*H* has order 2^{n}–1.

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 *I*_{0}(*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.