| import OAI.AlgebraicGeometry.CharacterVarieties.Frames.BandBases |
| import OAI.AlgebraicGeometry.CharacterVarieties.Cutting.RefinedDiagram |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace IntegralCharacterVarieties.SurfacePresentation.Diagram |
| open scoped Classical Matrix |
| open MatrixExpression NamedBandGrades OccurrenceIncidence |
| variable {F S V K : Type} {arity : S → ℕ} [Field K] |
| (D : Diagram F S V arity) (q : S) |
|
|
| /-- Connected-strip incidence constructed from a pair of seam frames. -/ |
| def cutFromOldFrames [Finite V] |
| (f h : (Matrix (Fin (D.seamDim q)) (Fin (D.seamDim q)) K)ˣ) := |
| D.refinedCutDiagram q (D.bandFromOldFrames q f h).shape rfl |
| (D.bandFromOldFrames q f h).rowRanks (D.bandFromOldFrames q f h).colRanks |
| end IntegralCharacterVarieties.SurfacePresentation.Diagram |
| end |
|
|
| noncomputable section |
| namespace IntegralCharacterVarieties.OccurrenceIncidence.PortWiring |
| open scoped Classical |
| open VertexTable |
| variable {F V : Type} {kind : V → Kind} (W : PortWiring kind) |
| (d : (v : V) → Decoration (kind v) F) (h : W.ColorMatch d) |
| lemma colored_local_color (x : LocalEnd V kind) : |
| (W.colored d h).facet (((W.colored d h).realize x).1)=(d x.1).color x.2 := by |
| change W.seamColor d (((W.assemble _ _).realize x).1)=_ |
| rw [W.realize_assemble] |
| exact W.seamColor_sideOf d h x |
| end IntegralCharacterVarieties.OccurrenceIncidence.PortWiring |
| namespace IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks |
| open scoped Classical |
| variable {R : Type} [CommRing R] {k : Kind} {d e : LocalRanks k} |
| def transportFrames (h : d=e) |
| (f : (p : k.table.Port) → MatrixIso R (d.Columns p) (d.Parent p)) : |
| (p : k.table.Port) → MatrixIso R (e.Columns p) (e.Parent p) := h ▸ f |
| lemma transportFrames_holds (h : d=e) |
| (f : (p : k.table.Port) → MatrixIso R (d.Columns p) (d.Parent p)) |
| (hf : (comparison k d f).Holds) : (comparison k e (transportFrames h f)).Holds := by |
| subst e |
| exact hf |
| end IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks |
| namespace IntegralCharacterVarieties.SurfacePresentation.Diagram |
| open scoped Classical |
| open OccurrenceIncidence VertexTable TwoFlagBand |
| variable {F S V : Type} {arity : S → ℕ} (D : Diagram F S V arity) (q : S) |
| variable {r : ℕ} (d : RankShape (arity q) (arity q) r) |
| (hp : D.rank (D.ports.facet ⟨q,none⟩)=r) |
| (hc : ∀ i,D.rank (D.ports.facet ⟨q,some i⟩)=d.secondaryRank (.row i)) |
| (hc' : ∀ i,D.rank (D.ports.facet ⟨q,some i⟩)=d.secondaryRank (.col i)) |
| variable [Finite V] |
| local notation "C" => D.refinedCutDiagram q d hp hc hc' |
| local notation "A" => D.ports.mapFacet (Sum.inl : F → D.ports.RefinedBandFacet q d) |
| local notation "B" => D.ports.refinedBandForSeam q d |
| lemma refinedCut_vertex_rank (v) (p) (c) : |
| ((C).vertexRanks v).rank p c=D.ports.refinedRank q d D.rank |
| ((BandGraft.decoration A q B v).color ⟨p,c⟩) := by |
| apply congrArg (D.ports.refinedRank q d D.rank) |
| ((BandGraft.wiring A q B).colored_local_color (BandGraft.decoration A q B) |
| (BandGraft.colorMatch A q B) ⟨v,p,c⟩) |
| lemma refinedCut_old_vertexRanks (v : V) : (C).vertexRanks (.inl v)=D.vertexRanks v := by |
| apply LocalRanks.ext_rank |
| funext p c |
| rw [D.refinedCut_vertex_rank q d hp hc hc'] |
| rfl |
| lemma refinedCut_orig_ranks (v : Fin (d.atomicBand.length+1)) (p) (c) : |
| (d.vertexAtoms v).rank ⟨p,c⟩=((C).vertexRanks (.inr (.inl v))).rank p c := by |
| exact ((d.vertexAtoms_rank v ⟨p,c⟩).trans |
| (D.ports.refinedBandForSeam_rank q d D.rank hp hc hc' v ⟨p,c⟩).symm).trans |
| (D.refinedCut_vertex_rank q d hp hc hc' (.inr (.inl v)) p c).symm |
| lemma refinedCut_mirror_ranks (v : Fin (d.atomicBand.length+1)) (p) (c) : |
| (d.vertexAtoms v).mirror.rank ⟨p,c⟩=((C).vertexRanks (.inr (.inr v))).rank p c := by |
| let z := (d.atomicBand.kind v).mirrorEnd.symm ⟨p,c⟩ |
| exact ((d.vertexAtoms_rank v z).trans |
| (D.ports.refinedBandForSeam_rank q d D.rank hp hc hc' v z).symm).trans |
| (D.refinedCut_vertex_rank q d hp hc hc' (.inr (.inr v)) p c).symm |
|
|
| variable {R : Type} [CommRing R] |
| def refinedOldFrames (f : D.PortFrames (R:=R)) (v : V) : |
| (p : D.ports.kind v |>.table.Port) → |
| @MatrixIso R _ (((C).vertexRanks (.inl v)).Columns p) (((C).vertexRanks (.inl v)).Parent p) |
| (((C).vertexRanks (.inl v)).fintypeColumns p) inferInstance := |
| LocalRanks.transportFrames (D.refinedCut_old_vertexRanks q d hp hc hc' v).symm |
| (fun p => f ⟨v,p⟩) |
|
|
| def refinedOrigFrames (v : Fin (d.atomicBand.length+1)) : |
| (p : (d.atomicBand.kind v).table.Port) → |
| @MatrixIso R _ (((C).vertexRanks (.inr (.inl v))).Columns p) |
| (((C).vertexRanks (.inr (.inl v))).Parent p) |
| (((C).vertexRanks (.inr (.inl v))).fintypeColumns p) inferInstance := |
| (d.vertexAtoms v).framesAtRanks _ (D.refinedCut_orig_ranks q d hp hc hc' v) |
|
|
| def refinedMirrorFrames (v : Fin (d.atomicBand.length+1)) : |
| (p : (d.atomicBand.kind v).mirror.table.Port) → |
| @MatrixIso R _ (((C).vertexRanks (.inr (.inr v))).Columns p) |
| (((C).vertexRanks (.inr (.inr v))).Parent p) |
| (((C).vertexRanks (.inr (.inr v))).fintypeColumns p) inferInstance := |
| (d.vertexAtoms v).mirror.framesAtRanks _ (D.refinedCut_mirror_ranks q d hp hc hc' v) |
|
|
| /-- The original and atomic frames satisfy the local vertex equations before the global gauge |
| changes. -/ |
| def refinedCutVertexFrames (f : D.PortFrames (R:=R)) (v : V ⊕ (Fin (d.atomicBand.length+1) ⊕ Fin (d.atomicBand.length+1))) : |
| (p : ((C).ports.kind v).table.Port) → |
| MatrixIso R (((C).vertexRanks v).Columns p) (((C).vertexRanks v).Parent p) := by |
| rcases v with v|v|v |
| · exact D.refinedOldFrames q d hp hc hc' f v |
| · exact D.refinedOrigFrames q d hp hc hc' v |
| · exact D.refinedMirrorFrames q d hp hc hc' v |
|
|
| def refinedCutPortFrames (f : D.PortFrames (R:=R)) : (C).PortFrames (R:=R) := |
| fun p => D.refinedCutVertexFrames q d hp hc hc' f p.1 p.2 |
|
|
| lemma refinedCutPortFrames_holds (f : D.PortFrames (R:=R)) |
| (hf : ∀ v,(LocalRanks.comparison (D.ports.kind v) (D.vertexRanks v) (fun p => f ⟨v,p⟩)).Holds) |
| (v) : (LocalRanks.comparison ((C).ports.kind v) ((C).vertexRanks v) |
| (fun p => D.refinedCutPortFrames q d hp hc hc' f ⟨v,p⟩)).Holds := by |
| rcases v with v|v|v |
| · change (LocalRanks.comparison _ _ (D.refinedOldFrames q d hp hc hc' f v)).Holds |
| exact LocalRanks.transportFrames_holds _ _ (hf v) |
| · exact (d.vertexAtoms v).framesAtRanks_holds _ (D.refinedCut_orig_ranks q d hp hc hc' v) |
| · exact (d.vertexAtoms v).mirror.framesAtRanks_holds _ (D.refinedCut_mirror_ranks q d hp hc hc' v) |
| end IntegralCharacterVarieties.SurfacePresentation.Diagram |
| end |
|
|
| end OAI |
|
|
