| import OAI.Analysis.GroupDeterminants.RegularOperators |
|
|
| namespace OAI |
|
|
| universe uF uR ur um |
|
|
| noncomputable section |
| open scoped BigOperators |
| |
| namespace GroupRingDeterminant |
|
|
| section ProjectionScalars |
| variable {F : Type uF} {R : Type uR} [Field F] [Ring R] [Algebra F R] |
|
|
| def projectionReweight (P : R) (a b : F) : R := a • P + b • (1 - P) |
|
|
| theorem projectionReweight_mul (P : R) (hP : P * P = P) (a b c d : F) : |
| projectionReweight P a b * projectionReweight P c d = |
| projectionReweight P (a * c) (b * d) := by |
| have h₁ : P * (1 - P) = 0 := by rw [mul_sub, mul_one, hP, sub_self] |
| have h₂ : (1 - P) * P = 0 := by rw [sub_mul, one_mul, hP, sub_self] |
| have h₃ : (1 - P) * (1 - P) = 1 - P := by |
| calc |
| (1 - P) * (1 - P) = 1 - P - P + P * P := by noncomm_ring |
| _ = 1 - P := by rw [hP] |
| simp only [projectionReweight, add_mul, mul_add, smul_mul_smul_comm, hP, |
| h₁, h₂, h₃, smul_zero, add_zero, zero_add] |
|
|
| @[simp] theorem projectionReweight_one (P : R) : projectionReweight P (1 : F) 1 = 1 := by |
| simp [projectionReweight] |
|
|
| def projectionReweightUnit (P : R) (hP : P * P = P) (a b : F) (ha : a ≠ 0) (hb : b ≠ 0) : |
| Rˣ where |
| val := projectionReweight P a b |
| inv := projectionReweight P a⁻¹ b⁻¹ |
| val_inv := by rw [projectionReweight_mul P hP] |
| inv_val := by rw [projectionReweight_mul P hP] |
|
|
| end ProjectionScalars |
|
|
| section ShearAlgebra |
| variable {R : Type uR} {r : Type ur} {m : Type um} [Ring R] [Fintype r] [Fintype m] [DecidableEq r] [DecidableEq m] |
|
|
| def lowerShear (S : Matrix m r R) : Matrix (r ⊕ m) (r ⊕ m) R := |
| Matrix.fromBlocks 1 0 S 1 |
|
|
| def upperShear (T : Matrix r m R) : Matrix (r ⊕ m) (r ⊕ m) R := |
| Matrix.fromBlocks 1 T 0 1 |
|
|
| theorem lowerShear_neg (S : Matrix m r R) : lowerShear S * lowerShear (-S) = 1 := by |
| simp [lowerShear, Matrix.fromBlocks_multiply, Matrix.fromBlocks_one] |
|
|
| theorem upperShear_neg (T : Matrix r m R) : upperShear T * upperShear (-T) = 1 := by |
| simp [upperShear, Matrix.fromBlocks_multiply, Matrix.fromBlocks_one] |
|
|
| def lowerShearUnit (S : Matrix m r R) : (Matrix (r ⊕ m) (r ⊕ m) R)ˣ where |
| val := lowerShear S |
| inv := lowerShear (-S) |
| val_inv := lowerShear_neg S |
| inv_val := by simpa using lowerShear_neg (-S) |
|
|
| def upperShearUnit (T : Matrix r m R) : (Matrix (r ⊕ m) (r ⊕ m) R)ˣ where |
| val := upperShear T |
| inv := upperShear (-T) |
| val_inv := upperShear_neg T |
| inv_val := by simpa using upperShear_neg (-T) |
|
|
| def blockDiagonalUnit (a : (Matrix r r R)ˣ) (b : (Matrix m m R)ˣ) : |
| (Matrix (r ⊕ m) (r ⊕ m) R)ˣ where |
| val := Matrix.fromBlocks ↑a 0 0 ↑b |
| inv := Matrix.fromBlocks ↑a⁻¹ 0 0 ↑b⁻¹ |
| val_inv := by simp [Matrix.fromBlocks_multiply, Matrix.fromBlocks_one] |
| inv_val := by simp [Matrix.fromBlocks_multiply, Matrix.fromBlocks_one] |
|
|
| end ShearAlgebra |
| |
| section TransferAlgebra |
| variable {F : Type uF} {R : Type uR} {r : Type ur} {m : Type um} [Field F] [Ring R] [Algebra F R] |
| variable [Fintype r] [Fintype m] [DecidableEq r] [DecidableEq m] |
|
|
| def transferMatrix (q : F) (P : Matrix m m R) (U : Matrix m r R) (V : Matrix r m R) : |
| Matrix (r ⊕ m) (r ⊕ m) R := |
| Matrix.fromBlocks (q • 1) V U (q⁻¹ • (U * V - P - q • (1 - P))) |
|
|
| theorem transferMatrix_factorization (q : F) (hq : q ≠ 0) (P : Matrix m m R) |
| (U : Matrix m r R) (V : Matrix r m R) : |
| lowerShear (q⁻¹ • U) * |
| Matrix.fromBlocks (q • (1 : Matrix r r R)) 0 0 |
| (projectionReweight P (-q⁻¹) (-1 : F)) * upperShear (q⁻¹ • V) = |
| transferMatrix q P U V := by |
| rw [lowerShear, upperShear, Matrix.fromBlocks_multiply, Matrix.fromBlocks_multiply] |
| simp only [Matrix.zero_mul, Matrix.mul_zero, Matrix.one_mul, Matrix.mul_one, |
| add_zero, zero_add, Matrix.smul_mul, Matrix.mul_smul, smul_smul] |
| simp only [inv_mul_cancel₀ hq, mul_inv_cancel₀ hq, one_smul] |
| apply Matrix.fromBlocks_inj.mpr |
| refine ⟨rfl, ?_, rfl, ?_⟩ |
| · simp |
| · simp only [projectionReweight, neg_smul, one_smul, smul_sub, smul_smul, |
| inv_mul_cancel₀ hq, mul_one, one_smul] |
| abel |
|
|
| theorem transferMatrix_isUnit (q : F) (hq : q ≠ 0) (P : Matrix m m R) |
| (hP : P * P = P) (U : Matrix m r R) (V : Matrix r m R) : |
| IsUnit (transferMatrix q P U V) := by |
| have hu : IsUnit (q • (1 : Matrix r r R)) := by |
| rw [Algebra.smul_def, mul_one] |
| exact (isUnit_iff_ne_zero.mpr hq).map (algebraMap F (Matrix r r R)) |
| obtain ⟨a, ha⟩ := hu |
| let b := projectionReweightUnit P hP (-q⁻¹) (-1 : F) |
| (neg_ne_zero.mpr (inv_ne_zero hq)) (neg_ne_zero.mpr one_ne_zero) |
| rw [← transferMatrix_factorization q hq P U V] |
| have hd : IsUnit (Matrix.fromBlocks (q • (1 : Matrix r r R)) 0 0 |
| (projectionReweight P (-q⁻¹) (-1 : F))) := by |
| refine ⟨blockDiagonalUnit a b, ?_⟩ |
| change Matrix.fromBlocks (a : Matrix r r R) 0 0 |
| (projectionReweight P (-q⁻¹) (-1 : F)) = _ |
| rw [ha] |
| exact (lowerShearUnit (q⁻¹ • U)).isUnit.mul hd |>.mul (upperShearUnit (q⁻¹ • V)).isUnit |
| |
| end TransferAlgebra |
| |
| end GroupRingDeterminant |
| |
| end |
| |
| end OAI |
| |
