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import OAI.Probability.BinarySweep.YoungTheory.YoungResolution
namespace OAI
noncomputable section
section
open scoped BigOperators Classical Kronecker
open Equiv Matrix
namespace BinaryCoordinateSweeps.Signed
variable {A : Type*} [Fintype A] [DecidableEq A] {n m : ℕ}
@[simp] lemma sumPerm_inv (g : Equiv.Perm (Fin n)) (h : Equiv.Perm (Fin m)) :
(sumPerm g h)⁻¹ = sumPerm g⁻¹ h⁻¹ := by
apply Equiv.ext
intro i
apply (sumPerm g h).injective
change (sumPerm g h) ((sumPerm g h).symm i) = _
rw [Equiv.apply_symm_apply]
induction i using Fin.addCases with
| left i => simp
| right i => simp
lemma signC_sum (β : Fin (n+m) → Bool)
(g : Equiv.Perm (Fin n)) (h : Equiv.Perm (Fin m)) :
signC β (sumPerm g h) =
signC (fun i => β (i.castAdd m)) g * signC (fun i => β (i.natAdd n)) h := by
simp only [signC,koszulSign_sum,Units.val_mul,Int.cast_mul]
def splitWord : (Fin (n+m) → A) ≃ (Fin n → A) × (Fin m → A) :=
(finSumFinEquiv.symm.arrowCongr (Equiv.refl A)).trans (Equiv.sumArrowEquivProdArrow _ _ _)
omit [Fintype A] [DecidableEq A] in
@[simp] lemma splitWord_fst (x : Fin (n+m) → A) (i : Fin n) :
(splitWord x).1 i = x (i.castAdd m) := rfl
omit [Fintype A] [DecidableEq A] in
@[simp] lemma splitWord_snd (x : Fin (n+m) → A) (i : Fin m) :
(splitWord x).2 i = x (i.natAdd n) := rfl
omit [Fintype A] [DecidableEq A] in
lemma splitWord_comp (x : Fin (n+m) → A) (g : Equiv.Perm (Fin n)) (h : Equiv.Perm (Fin m)) :
splitWord (x ∘ sumPerm g h) = ((splitWord x).1 ∘ g,(splitWord x).2 ∘ h) := by
apply Prod.ext <;> funext i <;> simp
lemma actionMatrix_apply (p : A → Bool) (g : Equiv.Perm (Fin n)) (x y : Fin n → A) :
LinearMap.toMatrix' (act p g) x y = if x ∘ g = y then signC (p ∘ x) g⁻¹ else 0 := by
simp only [LinearMap.toMatrix'_apply,act_apply,Pi.single_apply]
split_ifs with h
· simp
· simp
theorem actionMatrix_sum (p : A → Bool) (g : Equiv.Perm (Fin n)) (h : Equiv.Perm (Fin m)) :
LinearMap.toMatrix' (act p (sumPerm g h)) =
((LinearMap.toMatrix' (act p g)) ⊗ₖ (LinearMap.toMatrix' (act p h))).submatrix splitWord splitWord := by
ext x y
change LinearMap.toMatrix' (act p (sumPerm g h)) x y =
LinearMap.toMatrix' (act p g) (splitWord x).1 (splitWord y).1 *
LinearMap.toMatrix' (act p h) (splitWord x).2 (splitWord y).2
rw [actionMatrix_apply,actionMatrix_apply,actionMatrix_apply,sumPerm_inv,signC_sum]
have he : x ∘ sumPerm g h = y ↔
(splitWord x).1 ∘ g = (splitWord y).1 ∧ (splitWord x).2 ∘ h = (splitWord y).2 := by
rw [← splitWord.injective.eq_iff,splitWord_comp,Prod.mk.injEq]
simp only [he]
split_ifs <;> simp_all only [zero_mul,mul_zero,and_self,not_true_eq_false,true_and,false_and]; rfl
end BinaryCoordinateSweeps.Signed
end
open scoped BigOperators Classical
open Matrix
namespace BinaryCoordinateSweeps.Signed
open Density
variable {A : Type*} [Fintype A] [DecidableEq A] {n : ℕ}
abbrev actionMatrix (p : A → Bool) (g : Equiv.Perm (Fin n)) := LinearMap.toMatrix' (act p g)
lemma actionMatrix_conjugate_apply (p : A → Bool) (g : Equiv.Perm (Fin n))
(M : Matrix (Fin n → A) (Fin n → A) ℂ) (x y : Fin n → A) :
(actionMatrix p g * M * (actionMatrix p g).conjTranspose) x y=
signC (p ∘ x) g⁻¹ * signC (p ∘ y) g⁻¹ * M (x ∘ g) (y ∘ g) := by
simp only [Matrix.mul_apply,Matrix.conjTranspose_apply,actionMatrix_apply]
simp [apply_ite,signC_star,mul_assoc,mul_left_comm,mul_comm]
lemma actionMatrix_unitary (p : A → Bool) (g : Equiv.Perm (Fin n)) :
actionMatrix p g * (actionMatrix p g).conjTranspose=1 := by
have he (x y : Fin n → A) : x ∘ g = y ∘ g ↔ x=y :=
(wordReorder g).injective.eq_iff
ext x y
have hh := actionMatrix_conjugate_apply p g 1 x y
rw [mul_one] at hh
rw [hh]
by_cases h:x=y
· subst y
simp only [Matrix.one_apply,ite_true,signC_sq,mul_one]
· simp [he,h]
lemma actionMatrix_star_mul (p : A → Bool) (g : Equiv.Perm (Fin n)) :
(actionMatrix p g).conjTranspose * actionMatrix p g = 1 :=
mul_eq_one_comm.mp (actionMatrix_unitary p g)
theorem even_tensor_covariance (p : A → Bool) (g : Equiv.Perm (Fin n))
(M : Fin n → Matrix A A ℂ)
(hM : ∀ i a b, p a ≠ p b → M i a b=0) :
actionMatrix p g * tensorMatrices M * (actionMatrix p g).conjTranspose =
tensorMatrices (fun i => M (g⁻¹ i)) := by
ext x y
rw [actionMatrix_conjugate_apply]
have he : tensorMatrices M (x ∘ g) (y ∘ g) = tensorMatrices (fun i => M (g⁻¹ i)) x y := by
simpa [tensorMatrices] using
Equiv.prod_comp g (fun i => M (g⁻¹ i) (x i) (y i))
rw [he]
by_cases h : p ∘ x=p ∘ y
· rw [h,signC_sq,one_mul]
· obtain ⟨i,hi⟩ := Function.ne_iff.mp h
have hz : tensorMatrices (fun i => M (g⁻¹ i)) x y=0 :=
Finset.prod_eq_zero (Finset.mem_univ i) (hM _ _ _ hi)
rw [hz,mul_zero]
end BinaryCoordinateSweeps.Signed
end
end OAI
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