| import Mathlib |
| import OAI.Probability.BinarySweep.YoungTheory.YoungPlacement |
| import OAI.Probability.BinarySweep.Trajectories.TrajectoryComplement |
| import OAI.Probability.BinarySweep.Conditional.ComplementBlocks |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| open Irrep |
|
|
| variable {b h k : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h) |
| (x : Placement H k 0) (g₀ : ConditionalChoices H) |
|
|
| def augmentedOutputIndex : Placement H k 0 := |
| x.trans (remainingPerm H g₀.val g₀.property).toEmbedding |
|
|
| lemma augmentedOutputIndex_physical : |
| (augmentedPlacement H x g₀ (Fin.last b)).trans (freeIdentification H).symm.toEmbedding = |
| augmentedOutputIndex H x g₀ := by |
| apply Function.Embedding.ext |
| intro i |
| apply (freeIdentification H).injective |
| simp only [Function.Embedding.trans_apply,Equiv.toEmbedding_apply,Equiv.apply_symm_apply] |
| change augmentedPlacement H x g₀ (Fin.last b) i= |
| (freeIdentification H) ((freeIdentification H).symm (remainingBijection H g₀.val g₀.property (x i))) |
| rw [Equiv.apply_symm_apply] |
| apply Subtype.ext |
| exact congrArg (fun a : Equiv.Perm (GridSlot bits) => a (x i).val) (gridPartialSweep_last bits g₀.val) |
|
|
| def augmentationInputComplement : FreeSlot (augmentByChoice H x g₀) 0 ≃ PlacementComplement x := |
| (augmentationFreeEquiv H x g₀ 0).trans |
| (complementMap (Equiv.refl _) _ x (by |
| simpa only [Equiv.refl_toEmbedding,Function.Embedding.trans_refl] using augmentedPlacement_zero H x g₀)) |
|
|
| def augmentationOutputComplement : FreeSlot (augmentByChoice H x g₀) (Fin.last b) ≃ |
| PlacementComplement (augmentedOutputIndex H x g₀) := |
| (augmentationFreeEquiv H x g₀ (Fin.last b)).trans |
| (complementMap (freeIdentification H).symm _ _ (augmentedOutputIndex_physical H x g₀)) |
|
|
| lemma augmentationInputComplement_val (w : FreeSlot (augmentByChoice H x g₀) 0) : |
| (augmentationInputComplement H x g₀ w).val.val=w.val := rfl |
|
|
| lemma augmentationOutputComplement_val (w : FreeSlot (augmentByChoice H x g₀) (Fin.last b)) : |
| (augmentationOutputComplement H x g₀ w).val= |
| (freeIdentification H).symm (augmentationFreeEquiv H x g₀ (Fin.last b) w).val := rfl |
|
|
| lemma augmentedOutputIndex_event (g : ConditionalChoices H) : |
| x.trans (remainingPerm H g.val g.property).toEmbedding=augmentedOutputIndex H x g₀ ↔ |
| pathEvent (augmentByChoice H x g₀) g.val := by |
| rw [← placementEvent_iff_augmentation H x g₀ g] |
| constructor |
| · intro he |
| apply Function.Embedding.ext |
| intro i |
| apply (freeIdentification H).symm.injective |
| exact congrArg (fun y : Placement H k 0 => y i) he |
| · intro he |
| apply Function.Embedding.ext |
| intro i |
| exact congrArg (fun y : Placement H k (Fin.last b) => (freeIdentification H).symm (y i)) he |
|
|
| lemma augmented_event (g : ConditionalChoices (augmentByChoice H x g₀)) : |
| x.trans (remainingPerm H g.val ((augmentByChoice_extension H x g₀).pathEvent_mono g.property)).toEmbedding = |
| augmentedOutputIndex H x g₀ := |
| (augmentedOutputIndex_event H x g₀ ((augmentByChoice_extension H x g₀).restrictChoice g)).mpr g.property |
|
|
| lemma augmentation_complement_square (g : ConditionalChoices (augmentByChoice H x g₀)) |
| (w : FreeSlot (augmentByChoice H x g₀) 0) : |
| augmentationOutputComplement H x g₀ |
| (remainingBijection (augmentByChoice H x g₀) g.val g.property w) = |
| complementMap (remainingPerm H g.val ((augmentByChoice_extension H x g₀).pathEvent_mono g.property)) |
| x (augmentedOutputIndex H x g₀) |
| (augmented_event H x g₀ g) |
| (augmentationInputComplement H x g₀ w) := by |
| apply Subtype.ext |
| rw [augmentationOutputComplement_val,complementMap_val] |
| exact congrArg (freeIdentification H).symm (augmented_gridSweep H x g₀ g w) |
|
|
| variable {A : Type*} [Fintype A] (e : A ⊕ Fin k ≃ FreeSlot H 0) |
|
|
| def augmentedInputBasis : A ≃ FreeSlot (augmentByChoice H x g₀) 0 := |
| (placementComplementEquiv e x).trans (augmentationInputComplement H x g₀).symm |
|
|
| def augmentedOutputBasis : A ≃ FreeSlot (augmentByChoice H x g₀) (Fin.last b) := |
| (placementComplementEquiv e (augmentedOutputIndex H x g₀)).trans |
| (augmentationOutputComplement H x g₀).symm |
|
|
| lemma augmented_subgroup_bijection (g : ConditionalChoices (augmentByChoice H x g₀)) : |
| ((cosetEquiv (blockPerm e) (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e)).symm |
| ((placementSection e (augmentedOutputIndex H x g₀))⁻¹* |
| remainingPerm H g.val ((augmentByChoice_extension H x g₀).pathEvent_mono g.property))).1 = |
| ((augmentedInputBasis H x g₀ e).trans |
| (remainingBijection (augmentByChoice H x g₀) g.val g.property)).trans |
| (augmentedOutputBasis H x g₀ e).symm := by |
| rw [placement_subgroup_complement e |
| (remainingPerm H g.val ((augmentByChoice_extension H x g₀).pathEvent_mono g.property)) |
| x (augmentedOutputIndex H x g₀) (augmented_event H x g₀ g)] |
| ext a |
| apply (placementComplementEquiv e (augmentedOutputIndex H x g₀)).injective |
| simp only [Equiv.trans_apply,Equiv.apply_symm_apply,augmentedInputBasis,augmentedOutputBasis, |
| Equiv.symm_trans_apply] |
| exact (augmentation_complement_square H x g₀ g |
| ((augmentationInputComplement H x g₀).symm (placementComplementEquiv e x a))).symm.trans |
| (by rfl) |
|
|
| end BinaryCoordinateSweeps |
|
|
| end |
|
|
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps.Young |
| variable {I X : Type*} [Fintype I] [DecidableEq I] [Fintype X] [DecidableEq X] |
|
|
| def subPlacement (A : Finset I) (x : I ↪ X) : A ↪ X := |
| (Function.Embedding.subtype (fun i => i∈A)).trans x |
|
|
| omit [Fintype I] [DecidableEq I] [Fintype X] [DecidableEq X] in |
| lemma subPlacement_move (A : Finset I) (g : Equiv.Perm X) (x : I ↪ X) : |
| subPlacement A (movePlacement g x)=movePlacement g (subPlacement A x) := rfl |
|
|
| def placementMoveEquiv (g : Equiv.Perm X) : (I ↪ X) ≃ (I ↪ X) := |
| (Equiv.refl I).embeddingCongr g |
|
|
| omit [Fintype I] [DecidableEq I] [Fintype X] [DecidableEq X] in |
| lemma move_sub_eq_iff (A : Finset I) (g : Equiv.Perm X) |
| (x : I ↪ X) (u : A ↪ X) : |
| subPlacement A (movePlacement g x)=u ↔ subPlacement A x=movePlacement g⁻¹ u := by |
| rw [subPlacement_move] |
| constructor |
| · intro h |
| have he := congrArg (movePlacement g⁻¹) h |
| simpa only [← movePlacement_mul,inv_mul_cancel,movePlacement_one] using he |
| · intro h |
| rw [h,← movePlacement_mul,mul_inv_cancel,movePlacement_one] |
|
|
| def placementMarginal (A : Finset I) : ((I ↪ X) → ℂ) →ₗ[ℂ] ((A ↪ X) → ℂ) where |
| toFun v u := ∑x : I ↪ X, if subPlacement A x=u then v x else 0 |
| map_add' v w := by |
| funext u |
| simp only [Pi.add_apply] |
| rw [← Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro x _ |
| split_ifs <;> simp |
| map_smul' c v := by |
| funext u |
| simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply] |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro x _ |
| split_ifs <;> simp |
| |
| omit [DecidableEq I] in |
| lemma placementMarginal_intertwines (A : Finset I) (g : Equiv.Perm X) |
| (v : (I ↪ X) → ℂ) : |
| placementMarginal A (placementRep g v)=placementRep g (placementMarginal A v) := by |
| funext u |
| change (∑x : I ↪ X, if subPlacement A x=u then v (movePlacement g⁻¹ x) else 0) = |
| ∑x : I ↪ X, if subPlacement A x=movePlacement g⁻¹ u then v x else 0 |
| rw [← (placementMoveEquiv (I:=I) g).sum_comp |
| (fun x => if subPlacement A x=u then v (movePlacement g⁻¹ x) else 0)] |
| apply Finset.sum_congr rfl |
| intro x _ |
| change (if subPlacement A (movePlacement g x)=u then |
| v (movePlacement g⁻¹ (movePlacement g x)) else 0)=_ |
| simp only [move_sub_eq_iff,← movePlacement_mul,inv_mul_cancel,movePlacement_one] |
|
|
| variable (μ : YoungDiagram) |
|
|
| theorem spechtPlacement_marginal_zero (A : Finset (Tail μ)) |
| (hA : A≠Finset.univ) (v : SpechtSpace μ) : |
| placementMarginal A (spechtPlacement μ v)=0 := by |
| let f := (placementMarginal A).comp (spechtPlacement μ) |
| have hcard : Fintype.card A < Fintype.card (Cell μ) - μ.rowLen 0 := by |
| rw [← card_tail] |
| simp only [Fintype.card_coe] |
| exact Finset.card_lt_card (Finset.ssubset_univ_iff.mpr hA) |
| have hf : ∀g w, f (spechtRep μ g w)=placementRep g (f w) := by |
| intro g w |
| simp only [f,LinearMap.comp_apply,spechtPlacement_intertwines,placementMarginal_intertwines] |
| have he := no_short_placement_intertwiner μ hcard f hf |
| exact congrArg (fun f => f v) he |
|
|
| end BinaryCoordinateSweeps.Young |
|
|
| end |
|
|
| end OAI |
|
|
