openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /StrongDensePeriodicity.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.LinearGroups.StrongDenseEigenvalues | |
| import OAI.Probability.CriticalPercolation.LinearGroups.PeriodicSpectrum | |
| import OAI.Probability.CriticalPercolation.LinearGroups.ProjectivePeriodicity | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Module | |
| open scoped Classical | |
| namespace CriticalPercolation.LinearGrowth | |
| variable {K G n : Type} [Field K] [CharZero K] [Countable K] [IsAlgClosed K] | |
| variable [Group G] [DecidableEq G] [Fintype n] [DecidableEq n] [Nonempty n] | |
| variable {degree : ℕ} | |
| lemma scalar_power_of_strongFullMatrixSpan (ρ : G →* GL n K) | |
| (hρ : StrongFullMatrixSpan ρ) (hg : SparseGrowth.HasPolynomialDegree G degree) (g : G) : | |
| ∃ m : ℕ, 0 < m ∧ ∃ a : K, a≠0 ∧ (ρ g).val^m=a • (1:Matrix n n K) := by | |
| let A := (matrixRepresentation ρ g).toLinearMap | |
| obtain ⟨a,ha⟩ := Module.End.exists_eigenvalue A | |
| obtain ⟨v,hv⟩ := ha.exists_hasEigenvector | |
| have hav : (ρ g).val.mulVec v=a • v := hv.apply_eq_smul | |
| have ha0 : a≠0 := by | |
| intro he | |
| exact hv.2 ((matrixRepresentation ρ g).injective (by | |
| change (ρ g).val.mulVec v=(ρ g).val.mulVec 0 | |
| simpa only [he,zero_smul,Matrix.mulVec_zero] using hav)) | |
| obtain ⟨m,hm,hni⟩ := exists_scalar_unipotent_power A a ha0 (by | |
| intro r hr | |
| obtain ⟨w,hw⟩ := (Module.End.hasEigenvalue_of_isRoot hr).exists_hasEigenvector | |
| exact eigenvalue_ratio_root_of_unity_of_strongFullMatrixSpan ρ hρ hg g hw.2 hv.2 | |
| hw.apply_eq_smul hav) | |
| let : Uncountable ℂ := Arithmetic.uncountable_complete_normed_field ℂ | |
| obtain ⟨f'⟩ := Arithmetic.exists_algHom_of_countable (k:=ℚ) (K:=K) (F:=ℂ) | |
| let f := f'.toRingHom | |
| let τ := (Matrix.GeneralLinearGroup.map f).comp ρ | |
| have hτ := hρ.transfer f (RingHom.id ℂ) ρ τ (fun _ _ _ => rfl) | |
| let E : Matrix n n ℂ ≃ₐ[ℂ] ((n → ℂ) →L[ℂ] (n → ℂ)) := | |
| Matrix.toLinAlgEquiv'.trans continuousEndAlgEquiv | |
| let φ : Matrix n n K →+* ((n → ℂ) →L[ℂ] (n → ℂ)) := E.toRingHom.comp f.mapMatrix | |
| have hφs (c : K) : φ (c • (1:Matrix n n K))=f c • 1 := by | |
| have hmat : f.mapMatrix (c • (1:Matrix n n K))=f c • (1:Matrix n n ℂ) := by | |
| ext i j | |
| simp only [RingHom.mapMatrix_apply,Matrix.map_apply,Matrix.smul_apply,smul_eq_mul,map_mul] | |
| congr 1 | |
| exact congrArg (fun M : Matrix n n ℂ => M i j) f.mapMatrix.map_one | |
| change E (f.mapMatrix (c • (1:Matrix n n K)))=_ | |
| rw [hmat,map_smul,map_one] | |
| let T := φ (ρ g).val | |
| have hniM : IsNilpotent ((ρ g).val^m-a^m • (1:Matrix n n K)) := by | |
| have hh := hni.map (LinearMap.toMatrixAlgEquiv'.toRingHom) | |
| have hA : LinearMap.toMatrixAlgEquiv' A=(ρ g).val := | |
| Matrix.toLinAlgEquiv'.symm_apply_apply _ | |
| change IsNilpotent (LinearMap.toMatrixAlgEquiv' (A^m-a^m • 1)) at hh | |
| simpa only [map_sub,map_pow,map_smul,map_one,hA] using hh | |
| have hniT : IsNilpotent (T^m-(f a)^m • 1) := by | |
| have hh := hniM.map φ | |
| simpa only [map_sub,map_pow,hφs,T] using hh | |
| obtain ⟨S,h1,hi,hgen,hgS⟩ := (SparseGrowth.hasPolynomialDegree_iff degree).mp hg | |
| obtain ⟨δ,hδ,hb⟩ := projective_bound_of_strongFullSpan (matrixRepresentation τ) | |
| (hτ.strongFullSpan τ) S h1 hgen degree hgS | |
| have hTpow (k : ℕ) : (continuousRepresentation (matrixRepresentation τ) (g^k)).toContinuousLinearMap=T^k := by | |
| induction k with | |
| | zero => ext x i; simp only [pow_zero,map_one]; rfl | |
| | succ k ih => | |
| rw [pow_succ,map_mul,ContinuousLinearEquiv.toContinuousLinearMap_mul,ih,pow_succ] | |
| rfl | |
| have hbT (k : ℕ) (x : n → ℂ) : δ*‖T^k‖*‖x‖≤‖(T^k) x‖ := by | |
| rw [←hTpow] | |
| exact hb (g^k) x | |
| let v' : n → ℂ := f ∘ v | |
| have hv' : v'≠0 := eigenvector_transfer_ne_zero f (RingHom.id ℂ) v v' (fun _ => rfl) hv.2 | |
| have hav' : T v'=f a • v' := eigenvector_transfer f (RingHom.id ℂ) | |
| (ρ g).val (τ g).val (fun _ _ => rfl) v v' (fun _ => rfl) a (f a) rfl hav | |
| have hfa : f a≠0 := fun h => ha0 (f.injective (h.trans f.map_zero.symm)) | |
| have hs := scalar_power_of_projective_bound T δ hδ hbT hv' hfa hav' m hniT | |
| refine ⟨m,hm,a^m,pow_ne_zero m ha0,?_⟩ | |
| have hφinj : Function.Injective φ := by | |
| intro B C h | |
| have he := E.injective h | |
| ext i j | |
| apply f.injective | |
| exact congrArg (fun M : Matrix n n ℂ => M i j) he | |
| apply hφinj | |
| rw [map_pow,hφs,map_pow] | |
| exact hs | |
| end CriticalPercolation.LinearGrowth | |
| end | |
| end | |
| end OAI | |