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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 | 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
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