| import Mathlib |
| import OAI.Probability.BinarySweep.YoungTheory.HookInducedMoment |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps.Irrep |
| open Representation Equiv Equiv.Perm |
|
|
| variable {A B X : Type*} [Fintype A] [Fintype B] [Fintype X] |
| (e : A ⊕ B ≃ X) |
|
|
| abbrev PlacementComplement (x : B ↪ X) := {u : X // u ∉ Set.range x} |
|
|
| def placementComplementEquiv (x : B ↪ X) : A ≃ PlacementComplement x where |
| toFun a := ⟨placementSection e x (e (Sum.inl a)), by |
| rintro ⟨b,hb⟩ |
| have he := (placementSection e x).injective ((placementSection_apply e x b).trans hb) |
| exact Sum.inl_ne_inr (e.injective he).symm⟩ |
| invFun u := (e.symm ((placementSection e x)⁻¹ u.val)).getLeft (by |
| cases h : e.symm ((placementSection e x)⁻¹ u.val) with |
| | inl _ => rfl |
| | inr b => |
| exfalso |
| apply u.property |
| refine ⟨b,?_⟩ |
| have hh := congrArg (fun t => placementSection e x (e t)) h |
| simpa only [e.apply_symm_apply,Equiv.Perm.inv_def,Equiv.apply_symm_apply, |
| show placementSection e x (e (Sum.inr b)) = x b from placementSection_apply e x b] using hh.symm) |
| left_inv a := by |
| simp only [Equiv.Perm.inv_def,Equiv.symm_apply_apply,e.symm_apply_apply,Sum.getLeft_inl] |
| right_inv u := by |
| apply Subtype.ext |
| have hl : (e.symm ((placementSection e x)⁻¹ u.val)).isLeft := by |
| cases h : e.symm ((placementSection e x)⁻¹ u.val) with |
| | inl _ => rfl |
| | inr b => |
| exfalso |
| apply u.property |
| refine ⟨b,?_⟩ |
| have hh := congrArg (fun t => placementSection e x (e t)) h |
| simpa only [e.apply_symm_apply,Equiv.Perm.inv_def,Equiv.apply_symm_apply, |
| show placementSection e x (e (Sum.inr b)) = x b from placementSection_apply e x b] using hh.symm |
| change placementSection e x (e (Sum.inl ((e.symm ((placementSection e x)⁻¹ u.val)).getLeft _))) = u.val |
| rw [show Sum.inl ((e.symm ((placementSection e x)⁻¹ u.val)).getLeft hl) = |
| e.symm ((placementSection e x)⁻¹ u.val) from Sum.inl_getLeft _ hl] |
| simp |
|
|
| omit [Fintype X] in |
| lemma placementCoset_index (a : Equiv.Perm X) (y : B ↪ X) : |
| cosetIndexAction (blockPerm e) (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e) a y = y.trans a.symm.toEmbedding := by |
| let c := (cosetEquiv (blockPerm e) (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e)).symm ((placementSection e y)⁻¹*a) |
| have he : blockPerm e c.1 * (placementSection e c.2)⁻¹ = (placementSection e y)⁻¹*a := |
| (cosetEquiv (blockPerm e) (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e)).apply_symm_apply _ |
| have hi := congrArg (fun g : Equiv.Perm X => g⁻¹) he |
| ext b |
| have hb := Equiv.congr_fun hi (basePlacement e b) |
| simp only [mul_inv_rev,inv_inv,Equiv.Perm.mul_apply,← map_inv,placementSection_apply] at hb |
| change placementSection e c.2 ((blockPerm e c.1⁻¹) (e (Sum.inr b))) = _ at hb |
| rw [blockPerm_inr] at hb |
| exact (placementSection_apply e c.2 b).symm.trans hb |
|
|
| omit [Fintype X] in |
| lemma placementCoset_subgroup (a : Equiv.Perm X) (y : B ↪ X) : |
| blockPerm e ((cosetEquiv (blockPerm e) (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e)).symm ((placementSection e y)⁻¹*a)).1 = |
| (placementSection e y)⁻¹*a*placementSection e (y.trans a.symm.toEmbedding) := by |
| have he := (cosetEquiv (blockPerm e) (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e)).apply_symm_apply ((placementSection e y)⁻¹*a) |
| change blockPerm e _ * (placementSection e _)⁻¹ = _ at he |
| have hi := placementCoset_index e a y |
| change ((cosetEquiv (blockPerm e) (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e)).symm ((placementSection e y)⁻¹*a)).2 = _ at hi |
| rw [hi] at he |
| exact (mul_inv_eq_iff_eq_mul.mp he) |
|
|
| variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℂ W] |
| (σ : Representation ℂ (Equiv.Perm A) W) |
|
|
| omit [Fintype X] in |
| lemma placement_hilbertBlock (a : Equiv.Perm X) (x y : B ↪ X) : |
| hilbertBlock (coindHilbertRep (blockPerm e) σ (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e) a) y x = |
| if x.trans a.toEmbedding = y then |
| σ ((cosetEquiv (blockPerm e) (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e)).symm ((placementSection e y)⁻¹*a)).1 else 0 := by |
| rw [hilbertBlock_coind,placementCoset_index] |
| congr 1 |
| apply propext |
| constructor |
| · intro h |
| ext b |
| have hb := congrArg (fun x : B ↪ X => a (x b)) h |
| exact hb.symm.trans (by simp) |
| · intro h |
| ext b |
| have hb := congrArg (fun x : B ↪ X => a⁻¹ (x b)) h |
| simpa using hb.symm |
|
|
| end BinaryCoordinateSweeps.Irrep |
|
|
| end |
|
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| end OAI |
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|
