| import OAI.Analysis.GroupDeterminants.BinaryMatrices |
| |
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Pointwise |
|
|
| namespace GroupRingDeterminant |
| open LampDraft LampProjectionDraft |
| variable {H : Type} [Group H] |
|
|
| def lampRingEmbedding (R : Type) [Semiring R] : |
| MonoidAlgebra R H →+* MonoidAlgebra R (LampGroup H) := |
| MonoidAlgebra.mapDomainRingHom R (shift H) |
|
|
| lemma lampRingEmbedding_injective (R : Type) [Semiring R] : |
| Function.Injective (lampRingEmbedding (H := H) R) := by |
| exact MonoidAlgebra.mapDomain_injective (shift_injective H) |
|
|
| lemma binaryProjection_isolates {a : MonoidAlgebra (ZMod 2) H} {h : H} {S : Finset H} |
| (he : MonoidAlgebra.single h⁻¹ 1 * a = 1 + ∑ s ∈ S, MonoidAlgebra.single s 1) : |
| binaryProjection H S * LampProjectionDraft.shiftSymbol H h⁻¹ * |
| lampRingEmbedding (ZMod 2) a * binaryProjection H S = binaryProjection H S := by |
| have hmap := congrArg (lampRingEmbedding (H := H) (ZMod 2)) he |
| simp only [map_mul, map_add, map_one, map_sum] at hmap |
| have hn : LampProjectionDraft.shiftSymbol H h⁻¹ * lampRingEmbedding (ZMod 2) a = |
| 1 + ∑ s ∈ S, LampProjectionDraft.shiftSymbol H s := by |
| simpa [lampRingEmbedding, LampProjectionDraft.shiftSymbol] using hmap |
| rw [mul_assoc (binaryProjection H S), hn, mul_add, mul_one, add_mul, |
| (binaryProjection_idempotent H S).eq, Finset.mul_sum, Finset.sum_mul] |
| rw [Finset.sum_eq_zero (fun s hs => binaryProjection_separates H S hs), add_zero] |
| |
| lemma binary_defect_factorization_lamp {d : Type} [Fintype d] [DecidableEq d] |
| (X Y : Matrix d d (MonoidAlgebra (ZMod 2) H)) (hXY : X * Y = 1) (hYX : Y * X ≠ 1) : |
| ∃ S : Finset H, 1 ∉ S ∧ |
| ∃ u : Matrix (d ⊕ Unit) d (MonoidAlgebra (ZMod 2) (LampGroup H)), |
| ∃ v : Matrix d (d ⊕ Unit) (MonoidAlgebra (ZMod 2) (LampGroup H)), |
| u * v = Matrix.fromBlocks 1 0 0 (fun _ _ : Unit => binaryProjection H S) := by |
| classical |
| let C := 1 - Y * X |
| have hC : C ≠ 0 := sub_ne_zero.mpr (Ne.symm hYX) |
| obtain ⟨i, j, hij⟩ : ∃ i j, C i j ≠ 0 := by |
| by_contra! he |
| exact hC (Matrix.ext fun i j => he i j) |
| obtain ⟨h, S, hS, he⟩ := binary_coefficient_normalize (C i j) hij |
| let φ := lampRingEmbedding (H := H) (ZMod 2) |
| let Xm := X.map φ |
| let Ym := Y.map φ |
| let Cm := C.map φ |
| let p := binaryProjection H S |
| let f : Matrix Unit d (MonoidAlgebra (ZMod 2) (LampGroup H)) := |
| Matrix.single () i (p * LampProjectionDraft.shiftSymbol H h⁻¹) |
| let w : Matrix d Unit (MonoidAlgebra (ZMod 2) (LampGroup H)) := Matrix.single j () p |
| have hXYm : Xm * Ym = 1 := by |
| change X.map φ * Y.map φ = 1 |
| rw [← Matrix.map_mul, hXY, Matrix.map_one _ (map_zero _) (map_one _)] |
| have hCm : Cm = 1 - Ym * Xm := by |
| dsimp [Cm, C, Ym, Xm] |
| rw [Matrix.map_sub _ (map_sub _) _ _, Matrix.map_one _ (map_zero _) (map_one _), Matrix.map_mul] |
| have hfcw : f * Cm * w = fun _ _ : Unit => p := by |
| apply Matrix.ext |
| intro a b |
| cases a |
| dsimp only [f, w] |
| rw [Matrix.mul_single_apply_same, Matrix.single_mul_apply_same] |
| exact binaryProjection_isolates he |
| refine ⟨S, hS, Matrix.fromRows Xm (f * Cm), Matrix.fromCols Ym (Cm * w), ?_⟩ |
| rw [hCm, defect_factorization Xm Ym hXYm f w, ← hCm, hfcw] |
|
|
| end GroupRingDeterminant |
|
|
| end |
|
|
| end OAI |
|
|
