| import OAI.Probability.InvariantIsing.Haar.HaarPlaneFrame |
|
|
| namespace OAI |
|
|
| /-! The sum of squared Lie brackets, with the precise dimension factor. -/ |
| noncomputable section |
| open Matrix |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| def planeFunctionalMatrix {N : ℕ} |
| (ell : Matrix (Fin N) (Fin N) ℝ →ₗ[ℝ] ℝ) : Matrix (Fin N) (Fin N) ℝ := |
| Matrix.of fun i j => ell (planeGenerator i j) |
|
|
| lemma planeFunctionalMatrix_skew {N : ℕ} |
| (ell : Matrix (Fin N) (Fin N) ℝ →ₗ[ℝ] ℝ) : |
| (planeFunctionalMatrix ell).transpose = -planeFunctionalMatrix ell := by |
| ext i j |
| change ell (planeGenerator j i) = -ell (planeGenerator i j) |
| rw [planeGenerator_swap,map_neg] |
|
|
| lemma sum_plane_functional_bracket_sq {N : ℕ} |
| (ell : Matrix (Fin N) (Fin N) ℝ →ₗ[ℝ] ℝ) |
| (E : Matrix (Fin N) (Fin N) ℝ) (hE : E.transpose = -E) : |
| (∑ i, ∑ j, (ell (E*planeGenerator i j-planeGenerator i j*E))^2) = |
| matrixFrobeniusPair (E*planeFunctionalMatrix ell-planeFunctionalMatrix ell*E) |
| (E*planeFunctionalMatrix ell-planeFunctionalMatrix ell*E) := by |
| let A := planeFunctionalMatrix ell |
| have hA : A.transpose = -A := planeFunctionalMatrix_skew ell |
| have hs (i j : Fin N) : |
| 4*(ell (E*planeGenerator i j-planeGenerator i j*E))^2 = |
| (matrixFrobeniusPair (E*A-A*E) (planeGenerator i j))^2 := by |
| have hr := plane_functional_representation ell |
| (E*planeGenerator i j-planeGenerator i j*E) |
| (skew_matrix_bracket hE (planeGenerator_transpose i j)) |
| have hb := matrixFrobeniusPair_bracket E A (planeGenerator i j) hE |
| change matrixFrobeniusPair A _ = _ at hr |
| rw [hr] at hb |
| rw [hb] |
| ring |
| have ht := sum_matrixFrobeniusPair_plane_sq (E*A-A*E) (skew_matrix_bracket hE hA) |
| simp_rw [← hs] at ht |
| simp only [← Finset.mul_sum] at ht |
| change _ = matrixFrobeniusPair (E*A-A*E) (E*A-A*E) |
| linarith |
|
|
| /-- The ordered-plane convention has curvature factor `N - 2`. The factor |
| four here becomes one after the antisymmetric Hessian estimate. -/ |
| theorem sum_plane_functional_double_bracket_sq {N : ℕ} |
| (ell : Matrix (Fin N) (Fin N) ℝ →ₗ[ℝ] ℝ) : |
| (∑ i, ∑ j, ∑ k, ∑ l, |
| (ell (planeGenerator i j*planeGenerator k l- |
| planeGenerator k l*planeGenerator i j))^2) = |
| (4*((N : ℝ)-2))*(∑ i, ∑ j, (ell (planeGenerator i j))^2) := by |
| simp_rw [sum_plane_functional_bracket_sq ell _ (planeGenerator_transpose _ _)] |
| rw [sum_planeGenerator_bracket_norm_sq _ (planeFunctionalMatrix_skew ell), |
| matrixFrobeniusPair_self] |
| rfl |
|
|
| end InvariantIsing |
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|
| end |
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| end OAI |
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