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ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | 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)]
/-- Actual coefficients depend on the kernel, not on the chosen embedding of
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
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