| import OAI.Analysis.GroupDeterminants.Model |
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| namespace OAI |
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| universe uR ud ul um ur |
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| noncomputable section |
| open scoped BigOperators |
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| namespace GroupRingDeterminant |
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| def CompanionInput : Prop := |
| ∃ (K : Type) (_ : Field K) (_ : Finite K) (_ : CharP K 2) |
| (H : Type) (_ : Group H) (_ : Group.FG H) |
| (a b : MonoidAlgebra K H), a * b = 1 ∧ b * a ≠ 1 |
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| theorem no_finite_group_input (K H : Type) [Field K] [Finite K] [Group H] [Finite H] |
| (a b : MonoidAlgebra K H) (h : a * b = 1) : b * a = 1 := by |
| have : Finite (H →₀ K) := |
| Finite.of_injective (fun f : H →₀ K => (f : H → K)) DFunLike.coe_injective |
| have : Finite (MonoidAlgebra K H) := |
| Finite.of_equiv (H →₀ K) MonoidAlgebra.coeffEquiv.symm |
| exact mul_eq_one_symm h |
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| theorem regularOperator_unique {G : Type} [Group G] {n : ℕ} |
| {A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℤ G)} |
| {T S : Hilbert G n →L[ℂ] Hilbert G n} |
| (hT : IsRegularOperator A T) (hS : IsRegularOperator A S) : T = S := by |
| ext ξ ⟨i, h⟩ |
| exact (hT ξ i h).trans (hS ξ i h).symm |
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| theorem fkDet_pos {G : Type} [Group G] {n : ℕ} |
| (T : Hilbert G n →L[ℂ] Hilbert G n) : 0 < fkDet T := |
| Real.exp_pos _ |
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| section Defect |
| variable {R : Type uR} [Ring R] |
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| theorem defect_idempotent {x y : R} (h : x * y = 1) : |
| (1 - y * x) * (1 - y * x) = 1 - y * x := by |
| have hy : y * x * (y * x) = y * x := by |
| calc |
| y * x * (y * x) = y * (x * y) * x := by simp only [mul_assoc] |
| _ = y * x := by rw [h, mul_one] |
| calc |
| (1 - y * x) * (1 - y * x) = 1 - y * x - y * x + y * x * (y * x) := by |
| noncomm_ring |
| _ = 1 - y * x := by rw [hy]; noncomm_ring |
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| theorem mul_defect {x y : R} (h : x * y = 1) : x * (1 - y * x) = 0 := by |
| simp only [mul_sub, mul_one, ← mul_assoc, h, one_mul, sub_self] |
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| theorem defect_mul {x y : R} (h : x * y = 1) : (1 - y * x) * y = 0 := by |
| simp only [sub_mul, one_mul, mul_assoc, h, mul_one, sub_self] |
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| theorem defect_ne_zero {x y : R} (h : y * x ≠ 1) : 1 - y * x ≠ 0 := by |
| exact sub_ne_zero.mpr h.symm |
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| end Defect |
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| section MatrixDefect |
| variable {R : Type uR} {d : Type ud} {l : Type ul} [Ring R] [Fintype d] [DecidableEq d] |
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| theorem defect_factorization (X Y : Matrix d d R) (hXY : X * Y = 1) |
| (f : Matrix l d R) (w : Matrix d l R) : |
| Matrix.fromRows X (f * (1 - Y * X)) * Matrix.fromCols Y ((1 - Y * X) * w) = |
| Matrix.fromBlocks 1 0 0 (f * (1 - Y * X) * w) := by |
| rw [Matrix.fromRows_mul_fromCols, hXY] |
| congr 1 |
| · rw [← Matrix.mul_assoc, mul_defect hXY, Matrix.zero_mul] |
| · rw [Matrix.mul_assoc, defect_mul hXY, Matrix.mul_zero] |
| · rw [← Matrix.mul_assoc, Matrix.mul_assoc f, defect_idempotent hXY] |
|
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| end MatrixDefect |
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|
| section CornerLifting |
| variable {R : Type uR} [Ring R] |
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| def cornerSeries (e z : R) (k : ℕ) : R := |
| e * ∑ i ∈ Finset.range k, (e - z) ^ i |
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| theorem cornerSeries_spec (e z : R) (he : e * e = e) |
| (hez : e * z = z) (hze : z * e = z) (k : ℕ) (hk : (e - z) ^ k = 0) : |
| cornerSeries e z k * z = e ∧ z * cornerSeries e z k = e ∧ |
| e * cornerSeries e z k = cornerSeries e z k ∧ |
| cornerSeries e z k * e = cornerSeries e z k := by |
| let w := e - z |
| let s := ∑ i ∈ Finset.range k, w ^ i |
| have hew : e * w = w := by simp only [w, mul_sub, he, hez] |
| have hwe : w * e = w := by simp only [w, sub_mul, he, hze] |
| have hew' : Commute e w := hew.trans hwe.symm |
| have hes : Commute e s := |
| Commute.sum_right _ _ _ (fun i _ => hew'.pow_right i) |
| have hz : z = e * (1 - w) := by simp only [mul_sub, mul_one, hew, w]; noncomm_ring |
| have hinv : s * (1 - w) = 1 := by |
| simpa only [w, hk, sub_zero] using geom_sum_mul_neg w k |
| have hinv' : (1 - w) * s = 1 := by |
| simpa only [w, hk, sub_zero] using mul_neg_geom_sum w k |
| change (e * s) * z = e ∧ z * (e * s) = e ∧ |
| e * (e * s) = e * s ∧ (e * s) * e = e * s |
| have hq : (e * s) * e = e * s := by |
| rw [mul_assoc, ← hes.eq, ← mul_assoc, he] |
| refine ⟨?_, ?_, ?_, hq⟩ |
| · rw [hz, ← mul_assoc, hq, mul_assoc, hinv, mul_one] |
| · calc |
| z * (e * s) = z * s := by rw [← mul_assoc, hze] |
| _ = e := by rw [hz, mul_assoc, hinv', mul_one] |
| · rw [← mul_assoc, he] |
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| theorem lift_rectangular_factorization {m : Type um} {r : Type ur} [Fintype m] [DecidableEq m] |
| [Fintype r] (e : Matrix m m R) (he : e * e = e) |
| (u₀ : Matrix m r R) (v₀ : Matrix r m R) (k : ℕ) |
| (hk : (e - e * u₀ * v₀ * e) ^ k = 0) : |
| ∃ u : Matrix m r R, ∃ v : Matrix r m R, u * v = e := by |
| let z := e * u₀ * v₀ * e |
| have hez : e * z = z := by |
| dsimp [z] |
| rw [← Matrix.mul_assoc, ← Matrix.mul_assoc, ← Matrix.mul_assoc, he] |
| have hze : z * e = z := by |
| dsimp [z] |
| rw [Matrix.mul_assoc (e * u₀ * v₀), he] |
| obtain ⟨hqz, _, _, _⟩ := cornerSeries_spec e z he hez hze k hk |
| refine ⟨cornerSeries e z k * e * u₀, v₀ * e, ?_⟩ |
| simpa only [z, Matrix.mul_assoc] using hqz |
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| end CornerLifting |
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| end GroupRingDeterminant |
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| end |
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| end OAI |
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