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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 96 97 98 99 100 101 102 103 104 105 | import Mathlib
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical BigOperators
namespace CriticalPercolation.HomogeneousAnalysis
universe u v
variable {G : Type u} {X : Type v} [Group G] [MulAction G X]
lemma sum_smul_orbit (g : G) (x : X) (hf : (MulAction.orbit G x).Finite) (f : X → ℝ) :
(∑ y∈hf.toFinset, f (g • y))=∑ y∈hf.toFinset, f y := by
apply Finset.sum_bij (fun y _ => g • y)
· intro y hy
apply hf.mem_toFinset.mpr
obtain ⟨a,rfl⟩ := MulAction.mem_orbit_iff.mp (hf.mem_toFinset.mp hy)
exact ⟨g*a,mul_smul g a x⟩
· intro a _ b _ h
exact MulAction.injective g h
· intro y hy
refine ⟨g⁻¹ • y,hf.mem_toFinset.mpr ?_,smul_inv_smul g y⟩
obtain ⟨a,rfl⟩ := MulAction.mem_orbit_iff.mp (hf.mem_toFinset.mp hy)
exact ⟨g⁻¹*a,mul_smul g⁻¹ a x⟩
· intros; rfl
lemma finite_group_orbit_average [Fintype G] (x : X) (f : X → ℝ) :
(MulAction.orbit G x).ncard * (∑ g : G, f (g • x)) =
(Fintype.card G : ℝ) * ∑ y∈(Set.finite_range (fun g : G => g • x)).toFinset, f y := by
have hf : (MulAction.orbit G x).Finite := Set.finite_range (fun g : G => g • x)
have hinv (y : X) (hy : y∈hf.toFinset) :
(∑ g : G, f (g • y))=∑ g : G, f (g • x) := by
obtain ⟨a,rfl⟩ := MulAction.mem_orbit_iff.mp (hf.mem_toFinset.mp hy)
simp only [←mul_smul]
exact Equiv.sum_comp (Equiv.mulRight a) (fun g : G => f (g • x))
calc
_ = ∑ y∈hf.toFinset, ∑ g : G, f (g • y) := by
rw [Finset.sum_congr rfl hinv]
simp only [Finset.sum_const,nsmul_eq_mul,Set.ncard_eq_toFinset_card _ hf]
_ = ∑ g : G, ∑ y∈hf.toFinset, f (g • y) := Finset.sum_comm
_ = _ := by
simp only [sum_smul_orbit,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
rfl
def restrictedPermutation (B : Set X) (hB : ∀ g : G, ∀ x∈B, g • x∈B) (g : G) : Equiv.Perm B where
toFun x := ⟨g • x.val,hB g x.val x.property⟩
invFun x := ⟨g⁻¹ • x.val,hB g⁻¹ x.val x.property⟩
left_inv x := Subtype.ext (inv_smul_smul g x.val)
right_inv x := Subtype.ext (smul_inv_smul g x.val)
def restrictionHom (B : Set X) (hB : ∀ g : G, ∀ x∈B, g • x∈B) : G →* Equiv.Perm B where
toFun := restrictedPermutation B hB
map_one' := by ext x; exact one_smul G x.val
map_mul' g h := by ext x; exact mul_smul g h x.val
lemma exists_finite_restriction (B : Set X) (hfin : B.Finite)
(hB : ∀ g : G, ∀ x∈B, g • x∈B) :
∃ (H : Type v) (_ : Group H) (_ : Fintype H) (r : H → G),
(∀ h₁ h₂ x, x∈B → r (h₁*h₂) • x = r h₁ • (r h₂ • x)) ∧
(∀ g : G, ∃ h : H, ∀ x∈B, r h • x=g • x) := by
let := hfin.fintype
let τ := restrictionHom B hB
let H := τ.range
let : Fintype H := Fintype.ofFinite H
let r : H → G := fun h => Classical.choose h.property
have hr (h : H) : τ (r h)=h.val := Classical.choose_spec h.property
refine ⟨H,inferInstance,inferInstance,r,?_,?_⟩
· intro a b x hx
have he : τ (r (a*b))=τ (r a)*τ (r b) := by rw [hr,hr,hr]; rfl
exact congrArg (fun e : Equiv.Perm B => (e ⟨x,hx⟩).val) he
· intro g
refine ⟨⟨τ g,⟨g,rfl⟩⟩,?_⟩
intro x hx
exact congrArg (fun e : Equiv.Perm B => (e ⟨x,hx⟩).val) (hr ⟨τ g,⟨g,rfl⟩⟩)
lemma restricted_orbit_average {H : Type*} [Group H] [Fintype H]
(B : Set X) (r : H → G)
(hmul : ∀ a b x, x∈B → r (a*b) • x=r a • (r b • x))
(hsurj : ∀ g : G, ∃ h : H, ∀ x∈B, r h • x=g • x)
(x : X) (hx : x∈B) (hf : (MulAction.orbit G x).Finite) (f : X → ℝ) :
(MulAction.orbit G x).ncard * (∑ h : H, f (r h • x)) =
(Fintype.card H : ℝ) * ∑ y∈hf.toFinset, f y := by
have hinv (y : X) (hy : y∈hf.toFinset) :
(∑ h : H, f (r h • y))=∑ h : H, f (r h • x) := by
obtain ⟨g,rfl⟩ := MulAction.mem_orbit_iff.mp (hf.mem_toFinset.mp hy)
obtain ⟨a,ha⟩ := hsurj g
rw [←ha x hx]
simp_rw [←hmul _ _ x hx]
exact Equiv.sum_comp (Equiv.mulRight a) (fun h : H => f (r h • x))
calc
_ = ∑ y∈hf.toFinset, ∑ h : H, f (r h • y) := by
rw [Finset.sum_congr rfl hinv]
simp only [Finset.sum_const,nsmul_eq_mul,Set.ncard_eq_toFinset_card _ hf]
_ = ∑ h : H, ∑ y∈hf.toFinset, f (r h • y) := Finset.sum_comm
_ = _ := by
simp only [sum_smul_orbit,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
end CriticalPercolation.HomogeneousAnalysis
end
end
end OAI
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