| import Mathlib |
| import OAI.Probability.BinarySweep.YoungTheory.YoungClassification |
| import OAI.Probability.BinarySweep.Representations.MatrixIsotypicWhitening |
| import OAI.Probability.BinarySweep.Representations.IsotypicOrthogonal |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps.Young |
| open Representation |
|
|
| abbrev PartitionHilbert {n : ℕ} (p : n.Partition) := SpechtHilbert (diagram p) |
|
|
| def partitionHilbertRep {n : ℕ} (p : n.Partition) : |
| Representation ℂ (Equiv.Perm (Fin n)) (PartitionHilbert p) := |
| (hilbertSpecht (diagram p)).comp (cellsEquivFin p).symm.permCongrHom.toMonoidHom |
| |
| instance {n : ℕ} (p : n.Partition) : (partitionHilbertRep p).IsIrreducible := |
| Irrep.irreducible_comp_equiv _ _ |
| |
| def partitionHilbertEquiv {n : ℕ} (p : n.Partition) : |
| (partitionRep p).Equiv (partitionHilbertRep p) := |
| .mk (spechtHilbertEquiv (diagram p)) (by |
| intro g |
| apply LinearMap.ext |
| intro v |
| change spechtHilbertEquiv (diagram p) (spechtRep (diagram p) _ v) = |
| spechtHilbertEquiv (diagram p) (spechtRep (diagram p) _ |
| ((spechtHilbertEquiv (diagram p)).symm (spechtHilbertEquiv (diagram p) v))) |
| rw [LinearEquiv.symm_apply_apply]) |
| |
| lemma partitionHilbertRep_unitary {n : ℕ} (p : n.Partition) |
| (g : Equiv.Perm (Fin n)) (v : PartitionHilbert p) : |
| ‖partitionHilbertRep p g v‖=‖v‖ := hilbertSpecht_unitary _ _ _ |
| |
| lemma partitionHilbertRep_equiv_injective {n : ℕ} (p q : n.Partition) |
| (e : (partitionHilbertRep p).Equiv (partitionHilbertRep q)) : p=q := |
| partitionRep_equiv_injective p q |
| ((partitionHilbertEquiv p).trans (e.trans (partitionHilbertEquiv q).symm)) |
| |
| lemma partitionHilbertRep_complete {n : ℕ} {W : Type*} [AddCommGroup W] [Module ℂ W] |
| [FiniteDimensional ℂ W] (σ : Representation ℂ (Equiv.Perm (Fin n)) W) |
| [σ.IsIrreducible] : ∃ p : n.Partition, Nonempty (σ.Equiv (partitionHilbertRep p)) := by |
| obtain ⟨p,⟨e⟩⟩ := partitionRep_complete σ |
| exact ⟨p,⟨e.trans (partitionHilbertEquiv p)⟩⟩ |
| |
| lemma partition_isotypic_top {n : ℕ} {W : Type*} [AddCommGroup W] [Module ℂ W] |
| [FiniteDimensional ℂ W] (σ : Representation ℂ (Equiv.Perm (Fin n)) W) : |
| (⨆ p : n.Partition, Irrep.isotypicSpan (partitionHilbertRep p) σ)=⊤ := by |
| apply Irrep.iSup_isotypic_eq_top σ (fun p => partitionHilbertRep p) |
| intro U _ _ _ τ hτ |
| have := hτ |
| exact partitionHilbertRep_complete τ |
| |
| lemma partition_isotypic_orthogonal {n : ℕ} {W : Type*} |
| [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W] |
| (σ : Representation ℂ (Equiv.Perm (Fin n)) W) |
| (hσ : ∀ g v, ‖σ g v‖=‖v‖) : |
| OrthogonalFamily ℂ (fun p : n.Partition => Irrep.isotypicSpan (partitionHilbertRep p) σ) |
| (fun p => (Irrep.isotypicSpan (partitionHilbertRep p) σ).subtypeₗᵢ) := by |
| apply OrthogonalFamily.of_pairwise |
| intro p q hpq |
| apply Irrep.isotypic_orthogonal (partitionHilbertRep q) (partitionHilbertRep p) σ |
| (partitionHilbertRep_unitary q) (partitionHilbertRep_unitary p) hσ |
| rintro ⟨e⟩ |
| exact hpq (partitionHilbertRep_equiv_injective q p e).symm |
| |
| theorem partition_projection_sum {n : ℕ} {W : Type*} |
| [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W] |
| (σ : Representation ℂ (Equiv.Perm (Fin n)) W) |
| (hσ : ∀ g v, ‖σ g v‖=‖v‖) (v : W) : |
| (∑ p : n.Partition, (Irrep.isotypicSpan (partitionHilbertRep p) σ).starProjection v)=v := by |
| apply (partition_isotypic_orthogonal σ hσ).sum_projection_of_mem_iSup |
| rw [partition_isotypic_top] |
| trivial |
| |
| end BinaryCoordinateSweeps.Young |
| |
| end |
| |
| open scoped BigOperators Classical |
| open Equiv Equiv.Perm Finset |
| |
| namespace BinaryCoordinateSweeps.Signed |
| |
| def pairWeight {n : ℕ} (β : Fin n → Bool) (g : Equiv.Perm (Fin n)) |
| (a b : Fin n) : ℤˣ := |
| if b<a ∧ β a=true ∧ β b=true then (if g a ≤ g b then -1 else 1) else 1 |
| |
| lemma koszulSign_eq_prod {n : ℕ} (β : Fin n → Bool) (g : Equiv.Perm (Fin n)) : |
| koszulSign β g = ∏a, ∏b, pairWeight β g a b := by |
| rw [← Fintype.prod_sigma (fun x : Σ _ : Fin n, Fin n => pairWeight β g x.1 x.2)] |
| have he : finPairsLT n = Finset.univ.filter (fun x : Σ _ : Fin n, Fin n => x.2<x.1) := by |
| ext x |
| simp only [mem_finPairsLT,mem_filter,mem_univ,true_and] |
| rw [koszulSign,he,Finset.prod_filter] |
| apply Finset.prod_congr rfl |
| intro x _ |
| dsimp [pairWeight] |
| by_cases h : x.2<x.1 <;> by_cases h1 : β x.1=true <;> by_cases h2 : β x.2=true <;> simp [h,h1,h2] |
| |
| def sumPerm {n m : ℕ} (g : Equiv.Perm (Fin n)) (h : Equiv.Perm (Fin m)) : |
| Equiv.Perm (Fin (n+m)) := finSumFinEquiv.permCongr (Equiv.sumCongr g h) |
| |
| @[simp] lemma sumPerm_left {n m : ℕ} (g : Equiv.Perm (Fin n)) (h : Equiv.Perm (Fin m)) (i : Fin n) : |
| sumPerm g h (i.castAdd m) = (g i).castAdd m := by |
| change sumPerm g h (finSumFinEquiv (Sum.inl i)) = finSumFinEquiv (Sum.inl (g i)) |
| simp [sumPerm] |
| |
| @[simp] lemma sumPerm_right {n m : ℕ} (g : Equiv.Perm (Fin n)) (h : Equiv.Perm (Fin m)) (i : Fin m) : |
| sumPerm g h (i.natAdd n) = (h i).natAdd n := by |
| change sumPerm g h (finSumFinEquiv (Sum.inr i)) = finSumFinEquiv (Sum.inr (h i)) |
| simp [sumPerm] |
| |
| theorem koszulSign_sum {n m : ℕ} (β : Fin (n+m) → Bool) |
| (g : Equiv.Perm (Fin n)) (h : Equiv.Perm (Fin m)) : |
| koszulSign β (sumPerm g h) = |
| koszulSign (fun i => β (i.castAdd m)) g * koszulSign (fun i => β (i.natAdd n)) h := by |
| have hll (a b : Fin n) : pairWeight β (sumPerm g h) (a.castAdd m) (b.castAdd m) = |
| pairWeight (fun i => β (i.castAdd m)) g a b := by |
| simp only [pairWeight, sumPerm_left, Fin.lt_def, Fin.le_def, Fin.val_castAdd] |
| have hrr (a b : Fin m) : pairWeight β (sumPerm g h) (a.natAdd n) (b.natAdd n) = |
| pairWeight (fun i => β (i.natAdd n)) h a b := by |
| simp [pairWeight] |
| have hlr (a : Fin n) (b : Fin m) : pairWeight β (sumPerm g h) (a.castAdd m) (b.natAdd n) = 1 := by |
| simp [pairWeight,Fin.lt_def, show ¬ n+b.val<a.val by omega] |
| have hrl (a : Fin m) (b : Fin n) : pairWeight β (sumPerm g h) (a.natAdd n) (b.castAdd m) = 1 := by |
| simp [pairWeight,Fin.le_def,show ¬ n+(h a).val ≤ (g b).val by omega] |
| rw [koszulSign_eq_prod,Fin.prod_univ_add] |
| simp_rw [Fin.prod_univ_add,hll,hrr,hlr,hrl] |
| simp only [Finset.prod_const_one,mul_one,one_mul,← koszulSign_eq_prod] |
| |
| end BinaryCoordinateSweeps.Signed |
| |
| end |
| |
| end OAI |
| |
