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f205175 | 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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | 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
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