openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /InvariantSubspaceDensity.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.LinearGroups.FullSpanAlternative | |
| import OAI.Probability.CriticalPercolation.LinearGroups.BurnsideDensity | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Module | |
| open scoped Classical | |
| namespace CriticalPercolation.LinearGrowth | |
| variable {K V G : Type*} [Field K] [IsAlgClosed K] [AddCommGroup V] [Module K V] | |
| variable [FiniteDimensional K V] [Nontrivial V] [Group G] | |
| lemma invariant_subspace_of_not_fullSpan (ρ : G →* (V ≃ₗ[K] V)) (hρ : ¬FullSpan ρ) : | |
| ∃ W : Submodule K V, W≠⊥ ∧ W≠⊤ ∧ ∀ g, W.map (ρ g).toLinearMap=W := by | |
| let σ : G →* End K V := LinearEquiv.automorphismGroup.toLinearMapMonoidHom.comp ρ | |
| let A := Algebra.adjoin K ((MonoidHom.mrange σ) : Set (End K V)) | |
| have hAspan : A.toSubmodule=Submodule.span K (Set.range (fun g => (ρ g).toLinearMap)) := by | |
| rw [Algebra.adjoin_eq_span,Submonoid.closure_eq] | |
| rfl | |
| have hA : A≠⊤ := by | |
| intro he | |
| apply hρ | |
| change Submodule.span K (Set.range (fun g => (ρ g).toLinearMap))=⊤ | |
| rw [←hAspan,he] | |
| rfl | |
| have hns : ¬IsSimpleModule A V := by | |
| intro h | |
| exact hA (subalgebra_eq_top_of_simple A) | |
| have he : ∃ P : Submodule A V, P≠⊥ ∧ P≠⊤ := by | |
| by_contra hn | |
| push Not at hn | |
| have : IsSimpleOrder (Submodule A V) := IsSimpleOrder.of_forall_eq_top hn | |
| exact hns ⟨⟩ | |
| obtain ⟨P,hP0,hP1⟩ := he | |
| let W := P.restrictScalars K | |
| have hρmem (g : G) : (ρ g).toLinearMap∈A := | |
| Algebra.subset_adjoin (show σ g∈(MonoidHom.mrange σ) from ⟨g,rfl⟩) | |
| have hstable (g : G) {v : V} (hv : v∈W) : ρ g v∈W := | |
| P.smul_mem ⟨(ρ g).toLinearMap,hρmem g⟩ hv | |
| refine ⟨W,?_,?_,?_⟩ | |
| · intro hw | |
| apply hP0 | |
| apply Submodule.restrictScalars_injective K A V | |
| simpa only [Submodule.restrictScalars_bot] using hw | |
| · intro hw | |
| apply hP1 | |
| apply Submodule.restrictScalars_injective K A V | |
| simpa only [Submodule.restrictScalars_top] using hw | |
| · intro g | |
| ext v | |
| constructor | |
| · rintro ⟨x,hx,rfl⟩ | |
| exact hstable g hx | |
| · intro hv | |
| refine ⟨ρ g⁻¹ v,hstable g⁻¹ hv,?_⟩ | |
| simp only [map_inv,LinearEquiv.coe_coe,LinearEquiv.coe_inv, | |
| LinearEquiv.apply_symm_apply] | |
| end CriticalPercolation.LinearGrowth | |
| end | |
| end | |
| end OAI | |