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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 ''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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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