| import Mathlib |
| import OAI.Analysis.CoulombIonization.Variational.TensorLp |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
|
|
| open MeasureTheory Filter |
| open scoped BigOperators |
|
|
| namespace CoulombAtom |
|
|
| def blockPermHom (N M : ℕ) : |
| Equiv.Perm (Fin N) × Equiv.Perm (Fin M) →* Equiv.Perm (Fin (N+M)) := |
| finSumFinEquiv.permCongrHom.toMonoidHom.comp (Equiv.Perm.sumCongrHom (Fin N) (Fin M)) |
|
|
| def blockPerm {N M : ℕ} (π : Equiv.Perm (Fin N)) (ρ : Equiv.Perm (Fin M)) : |
| Equiv.Perm (Fin (N+M)) := blockPermHom N M (π,ρ) |
|
|
| def blockSubgroup (N M : ℕ) : Subgroup (Equiv.Perm (Fin (N+M))) := |
| (blockPermHom N M).range |
|
|
| lemma blockSubgroup_mem {N M : ℕ} {κ : Equiv.Perm (Fin (N+M))} : |
| κ ∈ blockSubgroup N M ↔ ∃ π ρ, blockPerm π ρ = κ := by |
| simp only [blockSubgroup, MonoidHom.mem_range, Prod.exists, blockPerm] |
|
|
| lemma blockPerm_apply_left {N M : ℕ} (π : Equiv.Perm (Fin N)) (ρ : Equiv.Perm (Fin M)) |
| (i : Fin N) : blockPerm π ρ (finSumFinEquiv (Sum.inl i)) = finSumFinEquiv (Sum.inl (π i)) := by |
| simp [blockPerm,blockPermHom,Equiv.permCongr,Equiv.Perm.sumCongrHom_apply] |
|
|
| lemma blockPerm_apply_right {N M : ℕ} (π : Equiv.Perm (Fin N)) (ρ : Equiv.Perm (Fin M)) |
| (i : Fin M) : blockPerm π ρ (finSumFinEquiv (Sum.inr i)) = finSumFinEquiv (Sum.inr (ρ i)) := by |
| simp [blockPerm,blockPermHom,Equiv.permCongr,Equiv.Perm.sumCongrHom_apply] |
|
|
| lemma blockPerm_sign {N M : ℕ} (π : Equiv.Perm (Fin N)) (ρ : Equiv.Perm (Fin M)) : |
| Equiv.Perm.sign (blockPerm π ρ) = Equiv.Perm.sign π * Equiv.Perm.sign ρ := by |
| simp [blockPerm,blockPermHom,Equiv.Perm.sign_sumCongr] |
|
|
| lemma blockSubgroup_mem_of_maps_left {N M : ℕ} (κ : Equiv.Perm (Fin (N+M))) |
| (h : ∀ i : Fin N, ∃ j : Fin N, |
| κ (finSumFinEquiv (Sum.inl i)) = finSumFinEquiv (Sum.inl j)) : |
| κ ∈ blockSubgroup N M := by |
| let σ : Equiv.Perm (Fin N ⊕ Fin M) := finSumFinEquiv.symm.permCongr κ |
| have hs : Set.MapsTo σ (Set.range Sum.inl) (Set.range Sum.inl) := by |
| rintro _ ⟨i,rfl⟩ |
| obtain ⟨j,hj⟩ := h i |
| refine ⟨j,?_⟩ |
| change Sum.inl j = finSumFinEquiv.symm (κ (finSumFinEquiv (Sum.inl i))) |
| rw [hj,Equiv.symm_apply_apply] |
| obtain ⟨⟨π,ρ⟩,he⟩ := Equiv.Perm.mem_sumCongrHom_range_of_perm_mapsTo_inl hs |
| apply blockSubgroup_mem.mpr |
| refine ⟨π,ρ,?_⟩ |
| apply finSumFinEquiv.symm.permCongr.injective |
| calc |
| _ = Equiv.Perm.sumCongrHom (Fin N) (Fin M) (π,ρ) := by |
| ext i |
| simp [blockPerm,blockPermHom,Equiv.permCongr] |
| _ = _ := he |
|
|
| lemma leftList_blockPerm {α : Type*} {N M : ℕ} (x : Fin (N+M) → α) |
| (π : Equiv.Perm (Fin N)) (ρ : Equiv.Perm (Fin M)) : |
| leftList (x ∘ blockPerm π ρ) = leftList x ∘ π := by |
| funext i |
| simp only [leftList, Function.comp_apply, blockPerm_apply_left] |
|
|
| lemma rightList_blockPerm {α : Type*} {N M : ℕ} (x : Fin (N+M) → α) |
| (π : Equiv.Perm (Fin N)) (ρ : Equiv.Perm (Fin M)) : |
| rightList (x ∘ blockPerm π ρ) = rightList x ∘ ρ := by |
| funext i |
| simp only [rightList, Function.comp_apply, blockPerm_apply_right] |
|
|
| end CoulombAtom |
|
|
| end |
|
|
| end OAI |
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|
