| import OAI.AlgebraicGeometry.CharacterVarieties.Frames.Graded |
| import OAI.AlgebraicGeometry.CharacterVarieties.Cutting.AtomicGallery |
|
|
| namespace OAI |
|
|
| namespace IntegralCharacterVarieties.NamedBandGrades |
| open scoped Classical |
| open TwoFlagBand OccurrenceIncidence.VertexTable |
| variable {K V α β : Type} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] |
| {n m : ℕ} |
|
|
| /-- A framed flag pair determines a finite atom-conservative band, with frames at every split, swap |
| and merge vertex and the original named endpoint quotient projections. -/ |
| theorem actual_named_atomic_band |
| (a : α → Fin n) (b : β → Fin m) |
| (f : (α → K) ≃ₗ[K] V) (h : (β → K) ≃ₗ[K] V) : |
| ∃ (d : RankShape n m (Module.finrank K V)) (e : (Atoms d → K) ≃ₗ[K] V) |
| (row : (k : ℕ) → ({i |
| (col : (k : ℕ) → ({i |
| (∀ k,framedPrefix (rowGrade d) e k=framedPrefix (fun i => (a i).val) f k) ∧ |
| (∀ k,framedPrefix (colGrade d) e k=framedPrefix (fun i => (b i).val) h k) ∧ |
| (∀ k v,v∈framedPrefix (rowGrade d) e (k+1) → |
| row k (framedGrade (rowGrade d) e k v)=framedGrade (fun i => (a i).val) f k v) ∧ |
| (∀ k v,v∈framedPrefix (colGrade d) e (k+1) → |
| col k (framedGrade (colGrade d) e k v)=framedGrade (fun i => (b i).val) h k v) ∧ |
| d.atomicBand.crossings.Nodup ∧ |
| (∀ v,∃ s : (d.atomicBand.kind v).SecondaryNames (Secondary n m), |
| d.atomicBand.decoration v=(d.atomicBand.kind v).freshDecoration s) ∧ |
| (∀ v,(d.atomicBand.decoration v).AtomConservative d.facetAtoms) ∧ |
| (∀ v z,(d.vertexAtoms v).rank z= |
| freshRank (Module.finrank K V) d.secondaryRank ((d.atomicBand.decoration v).color z)) ∧ |
| (∀ v,(LocalRanks.comparison (d.atomicBand.kind v) (d.vertexAtoms v).localRanks |
| ((d.vertexAtoms v).frame (R:=K))).Holds) := by |
| obtain ⟨d,e,row,col,hr,hc,hnr,hnc⟩ := actual_named_band_grades a b f h |
| exact ⟨d,e,row,col,hr,hc,hnr,hnc,d.atomicBand_nodup,d.atomicBand_fresh, |
| d.atomicBand_atoms,d.vertexAtoms_rank,d.atomicBand_comparison⟩ |
|
|
| end IntegralCharacterVarieties.NamedBandGrades |
|
|
| end OAI |
|
|
