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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | 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
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