5 - Coherence

Summary

(5.1) Definition. Let L and G be finite groups, A ⊂ L and S ⊂ Z[Irr L]. Let τ be a Z-linear isometry from E to Z[Irr G], where E is a Z-module such that Z[S,A] ⊂ E ⊂ Z[Irr L]. We say that (S, A, τ) is coherent, or that S is coherent, if Z[S, A] ≠ 0 and if there is a linear isometry from Z[S] to Z[Irr G] which coincides with τ on Z[S, A].

(5.2) Hypothesis, (a) Let L and G be finite groups and let S be a non-empty set of characters of L. Assume that, if χ ∈ S, then and.

(b) Assume that τ is a linear isometry fromZ[S, L^#] toZ[Irr G, G^#].

(c) The elements of S are pairwise orthogonal.

(d) Assume that, for for some orthonormal subset R(χ) ofZ[Irr G].

(e) If χ ∈ S, ϕ ∈ S and ϕ is orthogonal to then R(ϕ) is orthogonal to R(χ).

(5.3) (a) Assume (5.2.a), (5.2.b) and that S ⊂ Irr L. Then Hypothesis (5.2) holds.

(b) Assume Hypothesis (4.6), (5.2.a) and that

Then Hypothesis (5.2) holds with the isometry τ of Hypothesis (5.2) being the restriction toZ[S, L^#] of the isometry τ of Hypothesis (4.6).