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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | import Mathlib
namespace OAI
section
noncomputable section
open Set
open scoped Classical BigOperators
namespace ActionOrbitTransport
variable (Λ X : Type*) [Group Λ] [MulAction Λ X]
structure Representatives where
roots : Finset X
cover : ∀ x : X, ∃ u ∈ roots, ∃ φ : Λ, φ • u = x
unique : ∀ u ∈ roots, ∀ v ∈ roots, ∀ φ : Λ, φ • u = v → u = v
lemma finite_orbits (S : Set X) (hS : S.Finite)
(hcover : ∀ x : X, ∃ u∈S, x∈MulAction.orbit Λ u) :
Finite (MulAction.orbitRel.Quotient Λ X) := by
apply Set.finite_univ_iff.mp
apply (hS.image (fun x : X => (Quotient.mk'' x : MulAction.orbitRel.Quotient Λ X))).subset
intro q _
induction q using Quotient.inductionOn' with
| h x =>
obtain ⟨u,hu,φ,hφ⟩ := hcover x
refine ⟨u,hu,?_⟩
apply Quotient.sound
exact MulAction.mem_orbit_symm.mpr (MulAction.mem_orbit_iff.mpr ⟨φ,hφ⟩)
def representatives (S : Set X) (hS : S.Finite)
(hcover : ∀ x : X, ∃ u∈S, x∈MulAction.orbit Λ u) : Representatives Λ X := by
letI := finite_orbits Λ X S hS hcover
letI := Fintype.ofFinite (MulAction.orbitRel.Quotient Λ X)
let R : Finset X := Finset.univ.image (Quotient.out : MulAction.orbitRel.Quotient Λ X → X)
refine ⟨R,?_,?_⟩
· intro x
let q : MulAction.orbitRel.Quotient Λ X := Quotient.mk'' x
refine ⟨q.out,Finset.mem_image.mpr ⟨q,Finset.mem_univ _,rfl⟩,?_⟩
have heq : (Quotient.mk'' q.out : MulAction.orbitRel.Quotient Λ X) = Quotient.mk'' x := q.out_eq'
exact MulAction.mem_orbit_iff.mp (MulAction.mem_orbit_symm.mp (Quotient.exact heq))
· intro u hu v hv φ hφ
obtain ⟨q,_,rfl⟩ := Finset.mem_image.mp hu
obtain ⟨r,_,rfl⟩ := Finset.mem_image.mp hv
have heq : (Quotient.mk'' r.out : MulAction.orbitRel.Quotient Λ X) = Quotient.mk'' q.out :=
Quotient.sound (MulAction.mem_orbit_iff.mpr ⟨φ,hφ⟩)
rw [Quotient.out_eq',Quotient.out_eq'] at heq
exact congrArg Quotient.out heq.symm
abbrev PairOrbit := MulAction.orbitRel.Quotient Λ (X × X)
def pairOrbit (x y : X) : PairOrbit Λ X := Quotient.mk'' (x,y)
def stabilizerOrbit (x y : X) : Set X := MulAction.orbit (MulAction.stabilizer Λ x) y
def suborbitCard (x y : X) : ℕ := (stabilizerOrbit Λ X x y).ncard
lemma mem_stabilizerOrbit {x y z : X} : z ∈ stabilizerOrbit Λ X x y ↔
∃ φ : Λ, φ • x = x ∧ φ • y = z := by
simp only [stabilizerOrbit,MulAction.mem_orbit_iff]
constructor
· rintro ⟨φ,hφ⟩
exact ⟨φ,φ.property,hφ⟩
· rintro ⟨φ,hx,hy⟩
exact ⟨⟨φ,hx⟩,hy⟩
lemma pairOrbit_eq {x y u v : X} : pairOrbit Λ X x y = pairOrbit Λ X u v ↔
∃ φ : Λ, φ • x = u ∧ φ • y = v := by
constructor
· intro h
obtain ⟨φ,hφ⟩ := MulAction.mem_orbit_iff.mp (Quotient.exact h.symm)
exact ⟨φ,congrArg Prod.fst hφ,congrArg Prod.snd hφ⟩
· rintro ⟨φ,hx,hy⟩
apply Eq.symm
apply Quotient.sound
exact MulAction.mem_orbit_iff.mpr ⟨φ,Prod.ext hx hy⟩
lemma smul_pairOrbit (φ : Λ) (x y : X) :
pairOrbit Λ X (φ • x) (φ • y) = pairOrbit Λ X x y :=
(pairOrbit_eq Λ X |>.mpr ⟨φ,rfl,rfl⟩).symm
lemma smul_suborbitCard (φ : Λ) (x y : X) :
suborbitCard Λ X (φ • x) (φ • y) = suborbitCard Λ X x y := by
have heq : (fun z => φ • z) '' stabilizerOrbit Λ X x y =
stabilizerOrbit Λ X (φ • x) (φ • y) := by
ext z
constructor
· rintro ⟨w,hw,rfl⟩
obtain ⟨ψ,hx,hy⟩ := (mem_stabilizerOrbit Λ X).mp hw
apply (mem_stabilizerOrbit Λ X).mpr
refine ⟨φ*ψ*φ⁻¹,?_,?_⟩
· simp only [mul_smul,inv_smul_smul,hx]
· simp only [mul_smul,inv_smul_smul,hy]
· intro hz
obtain ⟨ψ,hx,hy⟩ := (mem_stabilizerOrbit Λ X).mp hz
refine ⟨φ⁻¹ • z,?_,smul_inv_smul φ z⟩
apply (mem_stabilizerOrbit Λ X).mpr
refine ⟨φ⁻¹*ψ*φ,?_,?_⟩
· simp only [mul_smul,hx,inv_smul_smul]
· simp only [mul_smul,hy]
rw [suborbitCard,←heq,Set.ncard_image_of_injective _ (MulAction.injective φ)]
rfl
private def swapFn : PairOrbit Λ X → PairOrbit Λ X :=
Quotient.map Prod.swap (by
intro a b hab
obtain ⟨φ,hφ⟩ := MulAction.mem_orbit_iff.mp hab
exact MulAction.mem_orbit_iff.mpr ⟨φ,Prod.ext (congrArg Prod.snd hφ) (congrArg Prod.fst hφ)⟩)
def swapPairOrbit : PairOrbit Λ X ≃ PairOrbit Λ X where
toFun := swapFn Λ X
invFun := swapFn Λ X
left_inv q := by induction q using Quotient.inductionOn' with | h p => rfl
right_inv q := by induction q using Quotient.inductionOn' with | h p => rfl
namespace Representatives
variable (R : Representatives Λ X)
abbrev RootedPair := {x // x∈R.roots} × X
def pairMap (p : R.RootedPair) : PairOrbit Λ X := pairOrbit Λ X p.1 p.2
lemma fiber_root_eq (x : {x // x∈R.roots}) (y : X)
(p : (pairMap Λ X R) ⁻¹' {pairOrbit Λ X x y}) : p.val.1 = x := by
have hh : pairOrbit Λ X (x:X) y = pairOrbit Λ X (p.val.1:X) p.val.2 := p.property.symm
obtain ⟨φ,hφ,_⟩ := (pairOrbit_eq Λ X).mp hh
exact Subtype.ext (R.unique x x.property p.val.1 p.val.1.property φ hφ).symm
lemma fiber_snd_mem (x : {x // x∈R.roots}) (y : X)
(p : (pairMap Λ X R) ⁻¹' {pairOrbit Λ X x y}) : p.val.2 ∈ stabilizerOrbit Λ X x y := by
have hh : pairOrbit Λ X (x:X) y = pairOrbit Λ X (p.val.1:X) p.val.2 := p.property.symm
obtain ⟨φ,hφ,hy⟩ := (pairOrbit_eq Λ X).mp hh
exact (mem_stabilizerOrbit Λ X).mpr ⟨φ,by rw [hφ,fiber_root_eq Λ X R x y p],hy⟩
def fiberEquiv (x : {x // x∈R.roots}) (y : X) :
(pairMap Λ X R) ⁻¹' {pairOrbit Λ X x y} ≃ stabilizerOrbit Λ X x y where
toFun p := ⟨p.val.2,fiber_snd_mem Λ X R x y p⟩
invFun z := ⟨(x,z),by
obtain ⟨φ,hx,hz⟩ := (mem_stabilizerOrbit Λ X).mp z.property
exact ((pairOrbit_eq Λ X).mpr ⟨φ,hx,hz⟩).symm⟩
left_inv p := by
apply Subtype.ext
exact Prod.ext (fiber_root_eq Λ X R x y p).symm rfl
right_inv z := rfl
end Representatives
end ActionOrbitTransport
end
end
end OAI
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