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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 | 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
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