| import OAI.AlgebraicGeometry.CharacterVarieties.Frames.FramedFlags |
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|
| namespace OAI |
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| 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 |
|
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| end OAI |
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