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a89d282 | 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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | 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
end OAI
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