| import Mathlib |
| import OAI.Computability.MaxCut.Encoding.KMSKernelOrbits |
| import OAI.Computability.MaxCut.Games.KMSBasisInvariant |
|
|
| namespace OAI |
|
|
| /-! Basis-invariant matrix functions have constant actual Fourier coefficients |
| on kernel fibers. This uses ambient automorphism transitivity proved from the |
| first isomorphism theorem, followed by the exact change-of-basis identity for |
| the normalized trace-character transform. -/ |
|
|
| namespace MaxCutGames.Inverse.KMSKernelOrbitsFourier |
|
|
| open MaxCutGames.Fourier.MatrixCharacters |
| open MaxCutGames.Fourier.MatrixFourier |
| open MaxCutGames.Inverse.KMSBasisInvariant |
| open MaxCutGames.Inverse.KMSKernelOrbits |
|
|
| noncomputable section |
|
|
| variable {E F : Type*} [AddCommGroup E] [Module F2 E] |
| [AddCommGroup F] [Module F2 F] |
| [FiniteDimensional F2 E] [Fintype (E →ₗ[F2] F)] |
|
|
| / |
| the quotient into the basis-index space. No coefficient-invariance premise is |
| assumed beyond the original function's basis invariance. -/ |
| theorem coefficient_eq_of_ker_eq (f : (E →ₗ[F2] F) → ℝ) |
| (hf : IsBasisInvariant f) (S T : F →ₗ[F2] E) (hker : S.ker = T.ker) : |
| linearCoeff f S = linearCoeff f T := by |
| obtain ⟨g, hg⟩ := exists_postcomp_equiv_of_ker_eq S T hker |
| have h := coefficient_change f hf g S |
| rw [hg] at h |
| exact h.symm |
|
|
| end |
| end MaxCutGames.Inverse.KMSKernelOrbitsFourier |
|
|
| end OAI |
| |
