| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem355.DiagonalStabilizer | |
| open Matrix | |
| variable {R : Type*} [CommRing R] | |
| def diagonal3 (D E : R) : Matrix (Fin 3) (Fin 3) R := | |
| Matrix.diagonal ![1, D, E] | |
| abbrev Ann (a : R) := {x : R // x * a = 0} | |
| abbrev FixingMatrices (D E : R) := | |
| {A : Matrix (Fin 3) (Fin 3) R // A * diagonal3 D E = diagonal3 D E} | |
| lemma fixing_entry {D E : R} (A : FixingMatrices D E) (i j : Fin 3) : | |
| A.val i j * ![1, D, E] j = if i = j then ![1, D, E] j else 0 := by | |
| have h := congrArg (fun M : Matrix (Fin 3) (Fin 3) R => M i j) A.property | |
| by_cases hij : i = j | |
| · simpa [diagonal3, Matrix.mul_diagonal, Matrix.diagonal_apply, hij] using h | |
| · simpa [diagonal3, Matrix.mul_diagonal, Matrix.diagonal_apply, hij] using h | |
| lemma fixing_first_column {D E : R} (A : FixingMatrices D E) (i : Fin 3) : | |
| A.val i 0 = if i = 0 then 1 else 0 := by | |
| simpa using fixing_entry A i 0 | |
| lemma fixing_other_column {D E : R} (A : FixingMatrices D E) (i j : Fin 3) : | |
| (A.val i j - if i = j then 1 else 0) * ![1, D, E] j = 0 := by | |
| have h := fixing_entry A i j | |
| by_cases hij : i = j | |
| · simpa [hij, sub_mul] using sub_eq_zero.mpr h | |
| · simpa [hij] using h | |
| def encode {D E : R} (A : FixingMatrices D E) : | |
| (Fin 3 → Ann D) × (Fin 3 → Ann E) := | |
| (fun i => ⟨A.val i 1 - if i = 1 then 1 else 0, | |
| by simpa using fixing_other_column A i 1⟩, | |
| fun i => ⟨A.val i 2 - if i = 2 then 1 else 0, | |
| by simpa using fixing_other_column A i 2⟩) | |
| lemma encode_injective {D E : R} : Function.Injective (encode (D := D) (E := E)) := by | |
| intro A A' h | |
| apply Subtype.ext | |
| apply Matrix.ext | |
| intro i j | |
| fin_cases j | |
| · change A.val i 0 = A'.val i 0 | |
| exact (fixing_first_column A i).trans (fixing_first_column A' i).symm | |
| · change A.val i 1 = A'.val i 1 | |
| have hv := congrArg (fun z => ((Prod.fst z) i).val) h | |
| simpa only [encode, sub_left_inj] using hv | |
| · change A.val i 2 = A'.val i 2 | |
| have hv := congrArg (fun z => ((Prod.snd z) i).val) h | |
| simpa only [encode, sub_left_inj] using hv | |
| theorem card_fixingMatrices_le [Finite R] (D E : R) : | |
| Nat.card (FixingMatrices D E) ≤ Nat.card (Ann D) ^ 3 * Nat.card (Ann E) ^ 3 := by | |
| simpa only [Nat.card_prod, Nat.card_fun, Nat.card_fin] using | |
| Nat.card_le_card_of_injective (encode (D := D) (E := E)) encode_injective | |
| abbrev FixingUnits (C : Matrix (Fin 3) (Fin 3) R) := | |
| {A : Matrix.GeneralLinearGroup (Fin 3) R // (A.val : Matrix (Fin 3) (Fin 3) R) * C = C} | |
| def forgetUnits {D E : R} (A : FixingUnits (diagonal3 D E)) : FixingMatrices D E := | |
| ⟨A.val.val, A.property⟩ | |
| lemma forgetUnits_injective {D E : R} : | |
| Function.Injective (forgetUnits (D := D) (E := E)) := by | |
| intro A A' h | |
| apply Subtype.ext | |
| apply Units.ext | |
| exact congrArg Subtype.val h | |
| theorem card_fixingUnits_diagonal_le [Finite R] (D E : R) : | |
| Nat.card (FixingUnits (diagonal3 D E)) ≤ | |
| Nat.card (Ann D) ^ 3 * Nat.card (Ann E) ^ 3 := by | |
| exact (Nat.card_le_card_of_injective forgetUnits forgetUnits_injective).trans | |
| (card_fixingMatrices_le D E) | |
| def conjugateFixing {C D : Matrix (Fin 3) (Fin 3) R} | |
| (P Q : Matrix.GeneralLinearGroup (Fin 3) R) | |
| (hC : C = (P : Matrix (Fin 3) (Fin 3) R) * D * (Q : Matrix (Fin 3) (Fin 3) R)) | |
| (A : FixingUnits C) : FixingUnits D := by | |
| refine ⟨P⁻¹ * A.val * P, ?_⟩ | |
| have h := A.property | |
| simp only [hC] at h | |
| have h' : (A.val.val * P.val * D) * Q.val = (P.val * D) * Q.val := by | |
| simpa only [Matrix.mul_assoc] using h | |
| have h'' : A.val.val * P.val * D = P.val * D := Q.isUnit.mul_right_cancel h' | |
| apply P.isUnit.mul_left_cancel | |
| simpa only [Units.val_mul, ← Matrix.mul_assoc, Units.mul_inv, one_mul] using h'' | |
| lemma conjugateFixing_injective {C D : Matrix (Fin 3) (Fin 3) R} | |
| (P Q : Matrix.GeneralLinearGroup (Fin 3) R) | |
| (hC : C = (P : Matrix (Fin 3) (Fin 3) R) * D * (Q : Matrix (Fin 3) (Fin 3) R)) : | |
| Function.Injective (conjugateFixing P Q hC) := by | |
| intro A A' h | |
| apply Subtype.ext | |
| have h' := congrArg Subtype.val h | |
| change P⁻¹ * A.val * P = P⁻¹ * A'.val * P at h' | |
| exact mul_left_cancel (mul_right_cancel h') | |
| theorem card_fixingUnits_of_diagonal_form_le [Finite R] | |
| (C : Matrix (Fin 3) (Fin 3) R) (D E : R) | |
| (P Q : Matrix.GeneralLinearGroup (Fin 3) R) | |
| (hC : C = (P : Matrix (Fin 3) (Fin 3) R) * diagonal3 D E * | |
| (Q : Matrix (Fin 3) (Fin 3) R)) : | |
| Nat.card (FixingUnits C) ≤ Nat.card (Ann D) ^ 3 * Nat.card (Ann E) ^ 3 := by | |
| exact (Nat.card_le_card_of_injective (conjugateFixing P Q hC) | |
| (conjugateFixing_injective P Q hC)).trans (card_fixingUnits_diagonal_le D E) | |
| theorem card_stabilizer_of_diagonal_form_le [Finite R] | |
| (C : Matrix (Fin 3) (Fin 3) R) (D E : R) | |
| (P Q : Matrix.GeneralLinearGroup (Fin 3) R) | |
| (hC : C = (P : Matrix (Fin 3) (Fin 3) R) * diagonal3 D E * | |
| (Q : Matrix (Fin 3) (Fin 3) R)) : | |
| Nat.card (MulAction.stabilizer (Matrix.GeneralLinearGroup (Fin 3) R) C) ≤ | |
| Nat.card (Ann D) ^ 3 * Nat.card (Ann E) ^ 3 := by | |
| change Nat.card (FixingUnits C) ≤ _ | |
| exact card_fixingUnits_of_diagonal_form_le C D E P Q hC | |
| end Problem355.DiagonalStabilizer | |
| end | |
| end OAI | |