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7d9acbd | 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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | 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
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