| import OAI.Probability.InvariantIsing.Haar.SpecialPlaneRotation |
|
|
| namespace OAI |
|
|
| /-! Compactness and entry bounds for the actual special orthogonal group. -/ |
| noncomputable section |
| open Matrix Set |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| lemma specialOrthogonal_entry_bound {N : ℕ} (U : SpecialOrthogonal N) (i j : Fin N) : |
| |(U : Matrix (Fin N) (Fin N) ℝ) i j| ≤ 1 := by |
| have hU := (Matrix.mem_orthogonalGroup_iff (Fin N) ℝ).mp |
| (Matrix.mem_specialOrthogonalGroup_iff.mp U.property).1 |
| have hrow : (∑ k, ((U : Matrix (Fin N) (Fin N) ℝ) i k)^2) = 1 := by |
| simpa only [Matrix.mul_apply,Matrix.transpose_apply,Matrix.one_apply_eq,pow_two] |
| using congrFun (congrFun hU i) i |
| have hs : ((U : Matrix (Fin N) (Fin N) ℝ) i j)^2 ≤ 1 := by |
| rw [← hrow] |
| exact Finset.single_le_sum (fun k _ => sq_nonneg _) (Finset.mem_univ j) |
| exact (sq_le_one_iff_abs_le_one _).mp hs |
|
|
| lemma isClosed_specialOrthogonal (N : ℕ) : |
| IsClosed (Matrix.specialOrthogonalGroup (Fin N) ℝ : Set (Matrix (Fin N) (Fin N) ℝ)) := by |
| have he : (Matrix.specialOrthogonalGroup (Fin N) ℝ : Set (Matrix (Fin N) (Fin N) ℝ)) = |
| {A | A*A.transpose=1} ∩ {A | A.det=1} := by |
| ext A |
| simp only [Set.mem_inter_iff,Set.mem_ofPred_eq,SetLike.mem_coe, |
| Matrix.mem_specialOrthogonalGroup_iff,Matrix.mem_orthogonalGroup_iff] |
| rw [he] |
| apply IsClosed.inter |
| · apply isClosed_eq <;> fun_prop |
| · apply isClosed_eq <;> fun_prop |
|
|
| lemma isCompact_specialOrthogonal (N : ℕ) : |
| IsCompact (Matrix.specialOrthogonalGroup (Fin N) ℝ : Set (Matrix (Fin N) (Fin N) ℝ)) := by |
| have hb : IsCompact (Set.pi Set.univ fun _ : Fin N => |
| Set.pi Set.univ fun _ : Fin N => Set.Icc (-1 : ℝ) 1) := |
| isCompact_univ_pi fun _ => isCompact_univ_pi fun _ => isCompact_Icc |
| apply hb.of_isClosed_subset (isClosed_specialOrthogonal N) |
| intro A hA |
| simp only [Set.mem_pi,Set.mem_univ,forall_true_left,Set.mem_Icc] |
| intro i j |
| exact abs_le.mp (specialOrthogonal_entry_bound ⟨A,hA⟩ i j) |
|
|
| instance specialOrthogonal_compactSpace (N : ℕ) : CompactSpace (SpecialOrthogonal N) := |
| isCompact_iff_compactSpace.mp (isCompact_specialOrthogonal N) |
|
|
| end InvariantIsing |
|
|
| end |
|
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| end OAI |
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|
