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7f5fab8 | 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 | import OAI.AlgebraicGeometry.CharacterVarieties.Foundation.Hurwitz
namespace OAI
noncomputable section
open scoped Classical Matrix
namespace IntegralCharacterVarieties.MatrixIso
open scoped Classical Matrix
variable {A : Type*} [CommRing A] {n : ℕ}
lemma unit_injective : Function.Injective (unit (R:=A) (α:=Fin n)) := by
intro x y h
apply Units.ext
exact congrArg MatrixIso.val h
end IntegralCharacterVarieties.MatrixIso
namespace IntegralCharacterVarieties.SurfacePresentation.Diagram
open scoped Classical Matrix
open HomTransport OccurrenceIncidence MatrixExpression
variable {F S V R A : Type} {arity : S → ℕ} [CommRing R] [CommRing A]
variable (D : Diagram F S V arity) (σ : ∀ f,Equiv.Perm (Fin (D.boundaryCount f)))
variable (φ : R →+* A)
variable (g : (e : D.Generator) → (Matrix (Fin (D.generatorRank e)) (Fin (D.generatorRank e)) A)ˣ)
/-- Puncture scalars and genus are not changed by the re-basing. -/
def reorderPunctures (P : D.Punctures R) : (D.reorderBoundary σ).Punctures R :=
⟨P.count,P.scalar⟩
lemma reorder_boundaryEdge_eval (f : F) (b : Fin (D.boundaryCount f))
(i : Fin (D.boundaryLength f (σ f b))) :
((D.reorderBoundary σ).boundaryEdgeWord f b i).eval φ (D.reorderGenerators σ g)=
(D.boundaryEdgeWord f (σ f b) i).eval φ g := by
apply MatrixIso.unit_injective
erw [(D.reorderBoundary σ).boundaryEdge_iso,D.boundaryEdge_iso]
rfl
lemma reorder_boundaryWord_eval (f : F) (b : Fin (D.boundaryCount f)) :
((D.reorderBoundary σ).boundaryWord f b).eval φ (D.reorderGenerators σ g)=
(D.boundaryWord f (σ f b)).eval φ g := by
erw [(D.reorderBoundary σ).boundaryWord_eval_raw,D.boundaryWord_eval_raw]
exact congrArg (fun x : Fin (D.boundaryLength f (σ f b)) →
(Matrix (Fin (D.rank f)) (Fin (D.rank f)) A)ˣ => (List.ofFn x).reverse.prod)
(funext fun i => D.reorder_boundaryEdge_eval σ φ g f b i)
lemma reorder_parentWord_eval (s : S) :
((D.reorderBoundary σ).parentWord s).eval φ (D.reorderGenerators σ g)=
(D.parentWord s).eval φ g := by
apply MatrixIso.unit_injective
unfold parentWord
erw [Term.eval_cast_iso φ (D.reorderGenerators σ g) ((D.reorderBoundary σ).seamRank s),
Term.eval_cast_iso φ g (D.seamRank s)]
rfl
lemma reorder_seamLeft_eval (s : S) :
((D.reorderBoundary σ).seamLeft s).eval φ (D.reorderGenerators σ g)=
(D.seamLeft s).eval φ g := by
simp only [seamLeft,Term.eval,D.reorder_parentWord_eval]
rfl
lemma reorder_seamRight_eval (s : S) :
((D.reorderBoundary σ).seamRight s).eval φ (D.reorderGenerators σ g)=
(D.seamRight s).eval φ g := rfl
lemma reorder_surfaceTarget_eval (P : D.Punctures R) (f : F) :
((D.reorderBoundary σ).surfaceTarget (D.reorderPunctures σ P) f).eval φ
(D.reorderGenerators σ g)=(D.surfaceTarget P f).eval φ g := rfl
lemma reorder_surfaceWord_eval (f : F) :
((D.reorderBoundary σ).surfaceWord f).eval φ (D.reorderGenerators σ g)=
(List.ofFn (fun k => (D.commutatorWord f k).eval φ g)).prod*
(List.ofFn (fun b => (D.boundaryWord f (σ f b)).eval φ g)).prod := by
simp only [surfaceWord,Term.eval,wordProduct_eval_raw,List.map_ofFn,Function.comp_def]
erw [show (fun b : Fin (D.boundaryCount f) =>
((D.reorderBoundary σ).boundaryWord f b).eval φ (D.reorderGenerators σ g)) =
(fun b => (D.boundaryWord f (σ f b)).eval φ g) from
funext (D.reorder_boundaryWord_eval σ φ g f)]
rfl
/-- The choices depend only on the finite permutation, not on a solution or on a coefficient field.
They are based path words. -/
def reorderPathWords (f : F) : Fin (D.boundaryCount f) → FreeGroup (Fin (D.boundaryCount f)) :=
(SurfaceSurgery.Hurwitz.finPresentation (σ f)).paths
lemma reorderPathWords_spec (f : F) {H : Type} [Group H]
(x : Fin (D.boundaryCount f) → H) :
(List.ofFn (fun b => (FreeGroup.lift x) (D.reorderPathWords σ f (σ f b))*x (σ f b)*
((FreeGroup.lift x) (D.reorderPathWords σ f (σ f b)))⁻¹)).prod=(List.ofFn x).prod :=
SurfaceSurgery.Hurwitz.finPresentation_spec (σ f) x
/-- Evaluate those same paths over an arbitrary coefficient ring. -/
def reorderCircleBases : D.CircleBases (R:=A) := fun ⟨f,b⟩ =>
MatrixIso.unit ((FreeGroup.lift (fun c => (D.boundaryWord f c).eval φ g))
(D.reorderPathWords σ f b))
def basedReorderGenerators := D.reorderGenerators σ
(D.gaugeGenerators (D.reorderCircleBases σ φ g) g)
lemma basedReorder_surfaceWord (f : F) :
((D.reorderBoundary σ).surfaceWord f).eval φ (D.basedReorderGenerators σ φ g)=
(D.surfaceWord f).eval φ g := by
rw [basedReorderGenerators,D.reorder_surfaceWord_eval]
simp only [D.boundaryWord_gauge,reorderCircleBases,MatrixIso.toUnit_unit]
rw [D.reorderPathWords_spec]
simp only [surfaceWord,Term.eval,wordProduct_eval_raw,List.map_ofFn]
rfl
variable [Algebra R A]
/-- Full sewn-diagram solution after arbitrary reordering of boundary circles. -/
def basedReorderSolution (P : D.Punctures R) (x : D.Solution P A) :
(D.reorderBoundary σ).Solution (D.reorderPunctures σ P) A := by
let G := D.reorderCircleBases σ (algebraMap R A) x.val.val
refine ⟨⟨D.basedReorderGenerators σ (algebraMap R A) x.val.val,?_⟩,?_⟩
· intro j
cases j with
| inl f =>
change SameFramedFlag (fun _ => 0)
(matrixUnitEquiv (((D.reorderBoundary σ).surfaceWord f).eval (algebraMap R A)
(D.basedReorderGenerators σ (algebraMap R A) x.val.val)))
(matrixUnitEquiv (((D.reorderBoundary σ).surfaceTarget (D.reorderPunctures σ P) f).eval
(algebraMap R A) (D.basedReorderGenerators σ (algebraMap R A) x.val.val)))
rw [D.basedReorder_surfaceWord]
exact x.val.property (.inl f)
| inr s =>
change SameFramedFlag (D.seamGrade s)
(matrixUnitEquiv (((D.reorderBoundary σ).seamLeft s).eval (algebraMap R A)
(D.reorderGenerators σ (D.gaugeGenerators G x.val.val))))
(matrixUnitEquiv (((D.reorderBoundary σ).seamRight s).eval (algebraMap R A)
(D.reorderGenerators σ (D.gaugeGenerators G x.val.val))))
rw [D.reorder_seamLeft_eval,D.reorder_seamRight_eval]
exact D.seamHolds_gauge (algebraMap R A) G x.val.val s (x.val.property (.inr s))
· intro v
change ((D.reorderBoundary σ).vertexComparison
(D.reorderGenerators σ (D.gaugeGenerators G x.val.val)) v).Holds
rw [D.reorder_vertexComparison]
exact D.vertexHolds_gauge G x.val.val x.property v
end IntegralCharacterVarieties.SurfacePresentation.Diagram
end
end OAI
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