File size: 3,372 Bytes
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 | import Mathlib
namespace OAI
section
noncomputable section
open Module Function
namespace CriticalPercolation.LinearGrowth
variable {K V G : Type*} [Field K] [AddCommGroup V] [Module K V] [Group G]
def subspaceRepresentation (ρ : G →* (V ≃ₗ[K] V)) (W : Submodule K V)
(hW : ∀ g, W.map (ρ g).toLinearMap=W) : G →* (W ≃ₗ[K] W) where
toFun g := (ρ g).ofSubmodules W W (hW g)
map_one' := by ext x; simp
map_mul' g h := by ext x; simp
def quotientRepresentation (ρ : G →* (V ≃ₗ[K] V)) (W : Submodule K V)
(hW : ∀ g, W.map (ρ g).toLinearMap=W) : G →* ((V ⧸ W) ≃ₗ[K] (V ⧸ W)) where
toFun g := Submodule.Quotient.equiv W W (ρ g) (hW g)
map_one' := by
ext x
induction x using Submodule.Quotient.induction_on
simp
map_mul' g h := by
ext x
induction x using Submodule.Quotient.induction_on
simp
@[simp] lemma subspaceRepresentation_apply (ρ : G →* (V ≃ₗ[K] V)) (W : Submodule K V)
(hW : ∀ g, W.map (ρ g).toLinearMap=W) (g : G) (w : W) :
(subspaceRepresentation ρ W hW g w : V)=ρ g w := rfl
@[simp] lemma quotientRepresentation_apply (ρ : G →* (V ≃ₗ[K] V)) (W : Submodule K V)
(hW : ∀ g, W.map (ρ g).toLinearMap=W) (g : G) (v : V) :
quotientRepresentation ρ W hW g (Submodule.Quotient.mk v)=Submodule.Quotient.mk (ρ g v) := rfl
lemma commute_of_identity_on_subspace_quotient (W : Submodule K V) (e f : V ≃ₗ[K] V)
(heW : ∀ w∈W, e w=w) (hfW : ∀ w∈W, f w=w)
(heQ : ∀ v, e v-v∈W) (hfQ : ∀ v, f v-v∈W) : e*f=f*e := by
ext v
have he := heW (f v-v) (hfQ v)
have hf := hfW (e v-v) (heQ v)
rw [map_sub] at he hf
change e (f v)=f (e v)
have h1 : e (f v)=f v-v+e v := (sub_eq_iff_eq_add).mp he
have h2 : f (e v)=e v-v+f v := (sub_eq_iff_eq_add).mp hf
rw [h1,h2]
abel
lemma solvable_of_subspace_quotient (ρ : G →* (V ≃ₗ[K] V)) (hρ : Injective ρ)
(W : Submodule K V) (hW : ∀ g, W.map (ρ g).toLinearMap=W)
[Group.IsSolvable (subspaceRepresentation ρ W hW).range]
[Group.IsSolvable (quotientRepresentation ρ W hW).range] : Group.IsSolvable G := by
let σ := subspaceRepresentation ρ W hW
let τ := quotientRepresentation ρ W hW
let ψ := σ.rangeRestrict.prod τ.rangeRestrict
have hker : Group.IsSolvable ψ.ker := by
apply Group.isSolvable_of_comm
intro x y
apply Subtype.ext
apply hρ
change ρ (x.val*y.val)=ρ (y.val*x.val)
rw [map_mul,map_mul]
have hs (g : ψ.ker) : σ g.val=1 := by
have he := congrArg (fun z : σ.range × τ.range => (z.1 : W ≃ₗ[K] W)) g.property
exact he
have ht (g : ψ.ker) : τ g.val=1 := by
have he := congrArg (fun z : σ.range × τ.range => (z.2 : (V ⧸ W) ≃ₗ[K] (V ⧸ W))) g.property
exact he
have hsub (g : ψ.ker) (w : V) (hw : w∈W) : ρ g.val w=w :=
congrArg (fun e : W ≃ₗ[K] W => (e ⟨w,hw⟩ : V)) (hs g)
have hquot (g : ψ.ker) (v : V) : ρ g.val v-v∈W := by
apply (Submodule.Quotient.eq W).mp
exact congrArg (fun e : (V ⧸ W) ≃ₗ[K] (V ⧸ W) => e (Submodule.Quotient.mk v)) (ht g)
exact commute_of_identity_on_subspace_quotient W (ρ x.val) (ρ y.val)
(hsub x) (hsub y) (hquot x) (hquot y)
let := hker
exact Group.isSolvable_of_ker_le_range ψ.ker.subtype ψ (by simp)
end CriticalPercolation.LinearGrowth
end
end
end OAI
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