openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /ProjectivelyPeriodicSchur.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.LinearGroups.SelbergSchur | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Function Module | |
| open scoped Classical | |
| namespace CriticalPercolation.LinearGrowth | |
| variable {K V G : Type*} [Field K] [CharZero K] [AddCommGroup V] [Module K V] | |
| variable [FiniteDimensional K V] [Group G] | |
| def linearMatrixEquiv : (V ≃ₗ[K] V) ≃* GL (Fin (finrank K V)) K := | |
| (LinearMap.GeneralLinearGroup.generalLinearEquiv K V).symm.trans | |
| (Units.mapEquiv (LinearMap.toMatrixAlgEquiv (Module.finBasis K V)).toMulEquiv) | |
| lemma finite_range_of_torsion_linearEquiv [Group.FG G] (ρ : G →* (V ≃ₗ[K] V)) | |
| (ht : ∀ g, IsOfFinOrder (ρ g)) : Finite ρ.range := by | |
| let τ := linearMatrixEquiv.toMonoidHom.comp ρ | |
| have ht' (g : G) : IsOfFinOrder (τ g) := by | |
| obtain ⟨m,hm,he⟩ := (ht g).exists_pow_eq_one | |
| apply isOfFinOrder_iff_pow_eq_one.mpr | |
| refine ⟨m,hm,?_⟩ | |
| change (linearMatrixEquiv (ρ g))^m=1 | |
| rw [←map_pow,he,map_one] | |
| let : Finite τ.range := finite_range_of_torsion τ ht' | |
| let f : ρ.range → τ.range := fun x => ⟨linearMatrixEquiv x.val,by | |
| obtain ⟨g,hg⟩ := x.property | |
| exact ⟨g,congrArg linearMatrixEquiv hg⟩⟩ | |
| exact Finite.of_injective f (fun x y h => Subtype.ext | |
| (linearMatrixEquiv.injective (congrArg Subtype.val h))) | |
| def conjugationRepresentation (ρ : G →* (V ≃ₗ[K] V)) : G →* (End K V ≃ₗ[K] End K V) where | |
| toFun g := (ρ g).conj | |
| map_one' := by ext f x; simp only [map_one]; rfl | |
| map_mul' g h := by ext f x; simp only [map_mul]; rfl | |
| omit [CharZero K] [FiniteDimensional K V] in | |
| lemma conjugation_eq_one_of_scalar (e : V ≃ₗ[K] V) (c : K) | |
| (hc : e.toLinearMap=c • (1:End K V)) : e.conj=1 := by | |
| have he (x : V) : e x=c • x := congrArg (fun f : End K V => f x) hc | |
| ext f x | |
| change e (f (e.symm x))=f x | |
| rw [he,←f.map_smul,←he,e.apply_symm_apply] | |
| lemma exists_finiteIndex_comm_range_of_projectively_periodic [Group.FG G] | |
| (ρ : G →* (V ≃ₗ[K] V)) | |
| (ht : ∀ g, ∃ m : ℕ, 0 < m ∧ ∃ c : K, | |
| ((ρ g).toLinearMap)^m=c • (1:End K V)) : | |
| ∃ H : Subgroup G, H.FiniteIndex ∧ | |
| ∀ x y : (ρ.comp H.subtype).range, x*y=y*x := by | |
| let τ := conjugationRepresentation ρ | |
| have htτ (g : G) : IsOfFinOrder (τ g) := by | |
| obtain ⟨m,hm,c,hc⟩ := ht g | |
| apply isOfFinOrder_iff_pow_eq_one.mpr | |
| refine ⟨m,hm,?_⟩ | |
| rw [←map_pow] | |
| apply conjugation_eq_one_of_scalar (ρ (g^m)) c | |
| rw [map_pow] | |
| change (LinearEquiv.automorphismGroup.toLinearMapMonoidHom (ρ g^m))=_ | |
| rw [map_pow] | |
| exact hc | |
| let : Finite τ.range := finite_range_of_torsion_linearEquiv τ htτ | |
| refine ⟨τ.ker,inferInstance,?_⟩ | |
| intro x y | |
| obtain ⟨g,hg⟩ := x.property | |
| obtain ⟨h,hh⟩ := y.property | |
| apply Subtype.ext | |
| change x.val*y.val=y.val*x.val | |
| rw [←hg,←hh] | |
| have hconj : (ρ g.val).conj=1 := g.property | |
| ext v | |
| have he := congrArg (fun e : End K V ≃ₗ[K] End K V => e (ρ h.val).toLinearMap (ρ g.val v)) hconj | |
| change (ρ g.val) ((ρ h.val) v)=(ρ h.val) ((ρ g.val) v) | |
| change (ρ g.val) ((ρ h.val) ((ρ g.val).symm ((ρ g.val) v)))=(ρ h.val) ((ρ g.val) v) at he | |
| simpa only [LinearEquiv.conj_apply_apply,LinearEquiv.symm_apply_apply, | |
| LinearEquiv.coe_one,id_eq,LinearEquiv.mul_apply,MonoidHom.comp_apply,Subgroup.coe_subtype] using he | |
| end CriticalPercolation.LinearGrowth | |
| end | |
| end | |
| end OAI | |