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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | 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
end OAI
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