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namespace OAI
noncomputable section
open scoped BigOperators
namespace GroupRingDeterminant
section ComplexRegular
variable {G : Type} [Group G] {n : ℕ}
def shiftFunctions : G →* Module.End ℂ (G → ℂ) where
toFun g := {
toFun f h := f (g⁻¹ * h)
map_add' _ _ := rfl
map_smul' _ _ := rfl }
map_one' := by ext f h; simp
map_mul' g h := by ext f a; simp [mul_assoc]
def convolutionHom : MonoidAlgebra ℂ G →ₐ[ℂ] Module.End ℂ (G → ℂ) :=
MonoidAlgebra.lift ℂ (Module.End ℂ (G → ℂ)) G shiftFunctions
theorem convolutionHom_apply (a : MonoidAlgebra ℂ G) (f : G → ℂ) (h : G) :
convolutionHom a f h = a.coeff.sum fun g z => z * f (g⁻¹ * h) := by
simp only [convolutionHom, MonoidAlgebra.lift_apply, Finsupp.sum,
LinearMap.sum_apply, LinearMap.smul_apply, Finset.sum_apply,
Pi.smul_apply, smul_eq_mul]
rfl
local instance functionModule : Module (MonoidAlgebra ℂ G) (G → ℂ) :=
Module.compHom (G → ℂ) convolutionHom.toRingHom
open scoped Matrix.Module
def complexOperator (A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G)) :
Hilbert G n →L[ℂ] Hilbert G n :=
∑ i : Fin n, ∑ j : Fin n, (A i j).coeff.sum
(fun g z => z • elementaryOperator i j g)
theorem complexOperator_apply (A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G))
(ξ : Hilbert G n) (i : Fin n) (h : G) :
complexOperator A ξ (i, h) = ∑ j : Fin n,
(A i j).coeff.sum fun g z => z * ξ (j, g⁻¹ * h) := by
classical
simp only [complexOperator, Finsupp.sum, sum_apply,
smul_apply, lp.coeFn_sum, Finset.sum_apply, lp.coeFn_smul,
Pi.smul_apply, smul_eq_mul, elementaryOperator_apply]
rw [Finset.sum_eq_single i]
· simp
· intro b _ hbi
simp [Ne.symm hbi]
· simp
theorem complexOperator_curry (A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G))
(ξ : Hilbert G n) :
Function.curry (complexOperator A ξ) = A • Function.curry (ξ : Fin n × G → ℂ) := by
funext i h
rw [Matrix.Module.smul_apply]
simp only [Finset.sum_apply]
rw [Function.curry_apply, complexOperator_apply]
apply Finset.sum_congr rfl
intro j _
exact (convolutionHom_apply (A i j) (fun g => ξ (j, g)) h).symm
theorem complexOperator_mul (A B : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G)) :
complexOperator (A * B) = complexOperator A * complexOperator B := by
apply ContinuousLinearMap.ext
intro ξ
apply lp.ext
funext a
obtain ⟨i, h⟩ := a
have he : Function.curry (complexOperator (A * B) ξ) =
Function.curry (complexOperator A (complexOperator B ξ)) := by
rw [complexOperator_curry, mul_smul, complexOperator_curry, complexOperator_curry]
exact congrFun (congrFun he i) h
theorem complexOperator_one : complexOperator (1 : Matrix (Fin n) (Fin n)
(MonoidAlgebra ℂ G)) = 1 := by
apply ContinuousLinearMap.ext
intro ξ
apply lp.ext
funext a
obtain ⟨i, h⟩ := a
have he := complexOperator_curry (1 : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G)) ξ
rw [one_smul] at he
exact congrFun (congrFun he i) h
theorem complexOperator_add (A B : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G)) :
complexOperator (A + B) = complexOperator A + complexOperator B := by
apply ContinuousLinearMap.ext
intro ξ
apply lp.ext
funext a
obtain ⟨i, h⟩ := a
have he : Function.curry (complexOperator (A + B) ξ) =
Function.curry (complexOperator A ξ) + Function.curry (complexOperator B ξ) := by
rw [complexOperator_curry, add_smul, complexOperator_curry, complexOperator_curry]
exact congrFun (congrFun he i) h
theorem complexOperator_zero : complexOperator (0 : Matrix (Fin n) (Fin n)
(MonoidAlgebra ℂ G)) = 0 := by
simp [complexOperator, MonoidAlgebra.coeff_zero]
def complexRepresentation : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G) →+*
(Hilbert G n →L[ℂ] Hilbert G n) where
toFun := complexOperator
map_one' := complexOperator_one
map_mul' := complexOperator_mul
map_zero' := complexOperator_zero
map_add' := complexOperator_add
theorem complexOperator_isUnit {A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G)}
(hA : IsUnit A) : IsUnit (complexOperator A) := hA.map complexRepresentation
theorem complexOperator_integer (A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℤ G)) :
complexOperator (A.map (MonoidAlgebra.mapRingHom G (Int.castRingHom ℂ))) =
regularOperator A := by
classical
unfold complexOperator regularOperator
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
change (MonoidAlgebra.map (Int.castRingHom ℂ).toAddMonoidHom (A i j)).coeff.sum _ = _
rw [MonoidAlgebra.coeff_map]
exact Finsupp.sum_mapRange_index
(h := fun groupElement coefficient => coefficient • elementaryOperator i j groupElement)
(fun groupElement => zero_smul ℂ (elementaryOperator i j groupElement))
theorem regularOperator_isUnit {A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℤ G)}
(hA : IsUnit (rationalMatrix A)) : IsUnit (regularOperator A) := by
let f : MonoidAlgebra ℚ G →+* MonoidAlgebra ℂ G :=
MonoidAlgebra.mapRingHom G (Rat.castHom ℂ)
have h := complexOperator_isUnit (hA.map f.mapMatrix)
have hf : f.comp (MonoidAlgebra.mapRingHom G (Int.castRingHom ℚ)) =
MonoidAlgebra.mapRingHom G (Int.castRingHom ℂ) := by
rw [← MonoidAlgebra.mapRingHom_comp]
congr 1
change IsUnit (complexOperator (A.map
(f.comp (MonoidAlgebra.mapRingHom G (Int.castRingHom ℚ))))) at h
rw [hf, complexOperator_integer] at h
exact h
def groupTraceCLM : (Hilbert G n →L[ℂ] Hilbert G n) →L[ℂ] ℂ :=
∑ i : Fin n, (innerSL ℂ (identityVector i)).comp
(ContinuousLinearMap.apply ℂ (Hilbert G n) (identityVector i))
theorem groupTraceCLM_apply (T : Hilbert G n →L[ℂ] Hilbert G n) :
groupTraceCLM T = groupTrace T := by
simp [groupTraceCLM, groupTrace]
theorem groupTrace_complex (A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G)) :
groupTrace (complexOperator A) = ∑ i : Fin n, (A i i).coeff 1 := by
classical
unfold groupTrace
apply Finset.sum_congr rfl
intro i _
have hi (ξ : Hilbert G n) : inner ℂ (identityVector i) ξ = ξ (i, 1) := by
simp [identityVector, lp.inner_single_left, RCLike.inner_apply]
rw [hi, complexOperator_apply]
rw [Finset.sum_eq_single i]
· rw [(A i i).coeff.sum_eq_single 1]
· simp [identityVector_apply]
· intro g _ hg
simp [identityVector_apply, hg]
· simp
· intro j _ hji
simp [identityVector_apply, hji]
· simp
theorem coefficient_mul_comm (a b : MonoidAlgebra ℂ G) :
(a * b).coeff 1 = (b * a).coeff 1 := by
rw [MonoidAlgebra.coeff_mul_apply_left a b, MonoidAlgebra.coeff_mul_apply_right b a]
simp only [mul_one, one_mul, mul_comm]
theorem groupTrace_complex_mul_comm
(A B : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G)) :
groupTrace (complexOperator A * complexOperator B) =
groupTrace (complexOperator B * complexOperator A) := by
classical
rw [← complexOperator_mul, ← complexOperator_mul, groupTrace_complex, groupTrace_complex]
simp only [Matrix.mul_apply, MonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
exact coefficient_mul_comm _ _
end ComplexRegular
end GroupRingDeterminant
end
end OAI
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