| import OAI.Analysis.GroupDeterminants.DefectProjections | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators Pointwise | |
| namespace GroupRingDeterminant | |
| open LampDraft LampProjectionDraft | |
| variable {H d : Type} [Group H] [Fintype d] [DecidableEq d] | |
| lemma lamp_projection_data (X Y : Matrix d d (MonoidAlgebra (ZMod 2) H)) | |
| (hXY : X * Y = 1) (hYX : Y * X ≠ 1) : | |
| ∃ (D : ℕ), Odd D ∧ ∃ P : Matrix (d ⊕ Unit) (d ⊕ Unit) (MonoidAlgebra ℚ (LampGroup H)), | |
| P * P = P ∧ IsSelfAdjoint P ∧ (Fintype.card d : ℚ) < ∑ i, (P i i).coeff 1 ∧ | |
| ∃ Z : Matrix (d ⊕ Unit) (d ⊕ Unit) (MonoidAlgebra ℤ (LampGroup H)), | |
| Z.map toRat = (D : ℚ) • P ∧ | |
| ∃ u : Matrix (d ⊕ Unit) d (MonoidAlgebra (ZMod 2) (LampGroup H)), | |
| ∃ v : Matrix d (d ⊕ Unit) (MonoidAlgebra (ZMod 2) (LampGroup H)), | |
| u * v = Z.map (toMod 2) := by | |
| classical | |
| obtain ⟨S, hS, u, v, huv⟩ := binary_defect_factorization_lamp X Y hXY hYX | |
| let p := projection H S | |
| let D := denominator H S | |
| let P : Matrix (d ⊕ Unit) (d ⊕ Unit) (MonoidAlgebra ℚ (LampGroup H)) := | |
| Matrix.fromBlocks 1 0 0 (Matrix.diagonal (fun _ : Unit => p)) | |
| let Z : Matrix (d ⊕ Unit) (d ⊕ Unit) (MonoidAlgebra ℤ (LampGroup H)) := | |
| Matrix.fromBlocks ((D : ℤ) • 1) 0 0 (Matrix.diagonal (fun _ : Unit => numerator H S)) | |
| have hp : p * p = p := (projection_idempotent H S).eq | |
| have hps : star p = p := (projection_selfAdjoint H S).star_eq | |
| refine ⟨D, denominator_odd H S, P, ?_, ?_, ?_, Z, ?_, u, v, ?_⟩ | |
| · dsimp only [P] | |
| rw [Matrix.fromBlocks_multiply] | |
| simp only [Matrix.one_mul, Matrix.mul_zero, Matrix.zero_mul, add_zero, zero_add, | |
| Matrix.diagonal_mul_diagonal, hp] | |
| · change Matrix.conjTranspose P = P | |
| dsimp only [P] | |
| rw [Matrix.fromBlocks_conjTranspose] | |
| simp only [Matrix.conjTranspose_one, Matrix.conjTranspose_zero, Matrix.diagonal_conjTranspose] | |
| congr 2 | |
| funext i | |
| exact hps | |
| · have he : (∑ i, (P i i).coeff 1) = (Fintype.card d : ℚ) + (p.coeff 1) := by | |
| simp [P, Fintype.sum_sum_type] | |
| rw [he, lt_add_iff_pos_right] | |
| rw [show p = projection H S from rfl, projection_trace H S hS] | |
| positivity | |
| · apply Matrix.ext | |
| intro i j | |
| rcases i with i | i <;> rcases j with j | j | |
| · change fullIntToRat H (((D : ℤ) • (1 : Matrix d d _)) i j) = | |
| (D : ℚ) • (1 : Matrix d d _) i j | |
| rw [show (((D : ℤ) • (1 : Matrix d d _)) i j) = (D : ℤ) • ((1 : Matrix d d _) i j) from rfl, | |
| fullIntToRat_zsmul] | |
| simp [Matrix.one_apply] | |
| · simp [Z, P, Matrix.map_apply] | |
| · simp [Z, P, Matrix.map_apply] | |
| · cases i; cases j | |
| exact numerator_toRat H S | |
| · rw [huv] | |
| apply Matrix.ext | |
| intro i j | |
| rcases i with i | i <;> rcases j with j | j | |
| · change (1 : Matrix d d _) i j = fullIntToBin H (((D : ℤ) • (1 : Matrix d d _)) i j) | |
| rw [show (((D : ℤ) • (1 : Matrix d d _)) i j) = (D : ℤ) • ((1 : Matrix d d _) i j) from rfl, | |
| fullIntToBin_zsmul] | |
| simp [Matrix.one_apply, (denominator_odd H S).natCast_zmod_two, D] | |
| · simp [Z, Matrix.map_apply, Matrix.fromBlocks] | |
| · simp [Z, Matrix.map_apply, Matrix.fromBlocks] | |
| · cases i; cases j | |
| rfl | |
| end GroupRingDeterminant | |
| end | |
| end OAI | |