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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]; noncomm_ring
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]; simp [ha, hb]
inv_val := by rw [projectionReweight_mul P hP]; simp [ha, hb]
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
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