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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 | import OAI.AlgebraicGeometry.CharacterVarieties.Frames.FramedFlags
namespace OAI
noncomputable section
namespace IntegralCharacterVarieties.SurfacePresentation.Diagram
open scoped Classical Matrix
open OccurrenceIncidence VertexTable MatrixExpression NamedBandGrades HomTransport
/-- Frame values depend only on the ports and ranks, hence are unchanged by the
choice of boundary enumeration. -/
lemma frameValues_canonical {F S V K : Type} {arity : S → ℕ}
[CommRing K] [Finite S] (D : Diagram F S V arity)
(frames : D.PortFrames (R:=K)) (s : S) (b : Bool) :
D.frameValues frames s b =
(SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).frameValues frames s b := rfl
/-- A frame generator evaluates in the canonical boundary presentation using the
same port frame. -/
lemma valuesFromPorts_frame_canonical {F S V K : Type} {arity : S → ℕ}
[CommRing K] [Finite S] (D : Diagram F S V arity)
(frames : D.PortFrames (R:=K)) (side : D.SideValues (R:=K))
(handle : D.HandleValues (R:=K)) (s : S) (b : Bool) :
D.valuesFromPorts frames side handle (.frame s b) =
(SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).frameValues frames s b := rfl
/-- Evaluating and reindexing a frame does not depend on the boundary enumeration. -/
lemma valuesFromPorts_frameCoordinates_canonical {F S V K α β : Type} {arity : S → ℕ}
[CommRing K] [Finite S] [Fintype α] [Fintype β] (D : Diagram F S V arity)
(frames : D.PortFrames (R:=K)) (side : D.SideValues (R:=K))
(handle : D.HandleValues (R:=K)) (s : S) (b : Bool)
(e : α ≃ Fin (D.seamDim s)) (f : β ≃ Fin (D.seamDim s)) :
((MatrixIso.unit (D.valuesFromPorts frames side handle (.frame s b))).reindex e f).linearEquiv =
((MatrixIso.unit ((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).frameValues
frames s b)).reindex e f).linearEquiv := rfl
/-- The evaluated parent coordinates use only the port and rank data. -/
lemma valuesFromPorts_parentCoordinates_canonical {F S V K R α : Type} {arity : S → ℕ}
[CommRing K] [CommRing R] [Finite S] [Fintype α] (D : Diagram F S V arity)
(frames : D.PortFrames (R:=K)) (side : D.SideValues (R:=K))
(handle : D.HandleValues (R:=K)) (s : S) (φ : R →+* K)
(e : α ≃ Fin (D.seamDim s)) :
((MatrixIso.unit ((D.parentWord s).eval φ (D.valuesFromPorts frames side handle))).reindex e e).linearEquiv =
((MatrixIso.unit (((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).parentWord s).eval φ
((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).valuesFromPorts
frames side handle))).reindex e e).linearEquiv := by
unfold parentWord
erw [Term.eval_cast_unit φ (D.valuesFromPorts frames side handle) (D.seamRank s), Term.eval_cast_unit φ ((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).valuesFromPorts frames side handle) (D.seamRank s)]
rfl
/-- The evaluated inverse child coordinates use only the port and rank data. -/
lemma valuesFromPorts_childCoordinates_canonical {F S V K R α : Type} {arity : S → ℕ}
[CommRing K] [CommRing R] [Finite S] [Fintype α] (D : Diagram F S V arity)
(frames : D.PortFrames (R:=K)) (side : D.SideValues (R:=K))
(handle : D.HandleValues (R:=K)) (s : S) (φ : R →+* K)
(e : α ≃ Fin (D.seamDim s)) :
((MatrixIso.unit ((Term.block (D.childDim s) (fun j => Term.inv (D.childWord s j))).eval φ
(D.valuesFromPorts frames side handle))).reindex e e).linearEquiv =
((MatrixIso.unit ((Term.block
((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).childDim s)
(fun j => Term.inv ((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).childWord s j))).eval φ
((SurfacePresentation.fromPorts D.ports D.rank D.genus D.seamRank).valuesFromPorts
frames side handle))).reindex e e).linearEquiv := rfl
/-- Child coordinates for values induced by ports and side transports. -/
lemma valuesFromPorts_childWord_coordinates
{F S V R K : Type} {arity : S → ℕ} [CommRing R] [Field K]
(D : Diagram F S V arity) (s : S) (φ : R →+* K)
(frames : D.PortFrames (R:=K)) (side : D.SideValues (R:=K))
(handle : D.HandleValues (R:=K)) {m : ℕ} (d : Fin m → ℕ)
(c : Fin m ≃ Fin (arity s)) (hc : ∀ i,d i=D.childDim s (c i))
(old : (i : Fin m) → (Matrix (Fin (d i)) (Fin (d i)) K)ˣ)
(ho : ∀ i,rebaseUnit (hc i).symm (side ⟨s,some (c i)⟩)=old i)
(e : ((i : Fin m) × Fin (d i)) ≃ Fin (D.seamDim s))
(he : e=(Equiv.sigmaCongr c (fun i=>finCongr (hc i))).trans
(blockIndex (D.childDim s)).symm) :
((MatrixIso.unit ((Term.block (D.childDim s)
(fun j => Term.inv (D.childWord s j))).eval φ
(D.valuesFromPorts frames side handle))).reindex e e).linearEquiv=
(MatrixIso.block (fun i=>MatrixIso.unit (old i)⁻¹)).linearEquiv := by
exact D.childWord_coordinates s φ (D.valuesFromPorts frames side handle)
d c hc old ho e he
end IntegralCharacterVarieties.SurfacePresentation.Diagram
end
end OAI
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