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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 | 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; cases b
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
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