| import Mathlib |
| import OAI.RepresentationTheory.PartialPermutation.Model |
|
|
| namespace OAI |
|
|
| section |
| open scoped Classical |
| open scoped BigOperators ComplexConjugate MonoidAlgebra |
| open scoped BigOperators ComplexConjugate |
| open scoped MonoidAlgebra BigOperators |
| open scoped BigOperators MonoidAlgebra Classical |
|
|
| attribute [local instance] Classical.propDecidable |
| open scoped MonoidAlgebra BigOperators |
| open scoped BigOperators |
| |
| namespace PartialPermutation |
|
|
| lemma blockSubgroup_commute {X : Type*} {M N : Set X} (h : Disjoint M N) |
| {g k : Equiv.Perm X} (hg : g ∈ blockSubgroup M) (hk : k ∈ blockSubgroup N) : |
| Commute g k := by |
| apply Equiv.Perm.Disjoint.commute |
| intro x |
| by_cases hx : x ∈ M |
| · right |
| exact hk x (fun hn => Set.disjoint_left.mp h hx hn) |
| · exact Or.inl (hg x hx) |
|
|
| lemma jointFactors_generate {G : Type*} [Group G] {b : ℕ} (H : Fin b → Subgroup G) : |
| ⨆ i, (H i).subgroupOf (⨆ j, H j) = ⊤ := by |
| apply Subgroup.map_subtype_inj.mp |
| rw [Subgroup.map_iSup] |
| simp only [Subgroup.map_subgroupOf_eq_of_le (le_iSup H _), |
| ← MonoidHom.range_eq_map, Subgroup.range_subtype] |
| |
| lemma jointFactors_commute {G : Type*} [Group G] {b : ℕ} (H : Fin b → Subgroup G) |
| (h : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k) : |
| ∀ i j, i ≠ j → ∀ g ∈ (H i).subgroupOf (⨆ j, H j), |
| ∀ k ∈ (H j).subgroupOf (⨆ j, H j), Commute g k := by |
| intro i j hij g hg k hk |
| apply Subtype.ext |
| exact h i j hij g hg k hk |
|
|
| end PartialPermutation |
|
|
| end |
|
|
| end OAI |
|
|
