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607ee45 | 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 | import OAI.RepresentationTheory.RowColumn.CopyOverlap
import OAI.RepresentationTheory.RowColumn.Sectors
namespace OAI
section
noncomputable section
open scoped BigOperators Classical MatrixOrder Matrix.Norms.L2Operator ComplexOrder
namespace RowColumn
open CubeShuffle.Specht CubeShuffle.UnitaryFinite Signed MatrixState
theorem exists_hook_isometry (a : YoungDiagram) (h : ℕ) (H : InHook a h) :
∃ K : hilbertSpace a →ₗᵢ[ℂ] Sector (S := Cell a) (HookModel.IsOdd (h := h)) (HookModel.oddCount a h H),
∀ g x, K (unitaryRepresentation a g x) = HookModel.hookRepresentation a h H g (K x) := by
let := unitaryRepresentation_irreducible a
let f : Representation.IntertwiningMap (unitaryRepresentation a) (HookModel.hookRepresentation a h H) :=
(HookModel.hookEmbedding a h H).comp (unitaryEquiv a).symm.toIntertwiningMap
have hf : Function.Injective f := by
intro x y hxy
apply (spaceHilbertEquiv a).symm.injective
apply HookModel.hookEmbedding_injective a h H
exact hxy
have hu : IsUnitary (V := Sector (S := Cell a) (HookModel.IsOdd (h := h)) (HookModel.oddCount a h H)) (HookModel.hookRepresentation a h H) :=
Signed.representation_unitary (S := Cell a) (odd := HookModel.IsOdd (h := h)) (k := HookModel.oddCount a h H)
have hh := @isometric_intertwiner_of_injective
(Equiv.Perm (Cell a)) (hilbertSpace a)
(Sector (S := Cell a) (HookModel.IsOdd (h := h)) (HookModel.oddCount a h H))
inferInstance inferInstance inferInstance inferInstance inferInstance inferInstance inferInstance
(unitaryRepresentation a) (HookModel.hookRepresentation a h H)
(unitaryRepresentation_unitary a) hu (unitaryRepresentation_irreducible a) f hf
exact hh
section LineCopy
variable {S L V : Type*} [Fintype S] [DecidableEq S] [Fintype L] [DecidableEq L]
[NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
{u : ℕ} (a : YoungDiagram) (e : S ≃ Cell a) (h : ℕ) (H : InHook a h)
(K : hilbertSpace a →ₗᵢ[ℂ] Sector (S := Cell a) (HookModel.IsOdd (h := h)) (HookModel.oddCount a h H))
(hK : ∀ g x, K (unitaryRepresentation a g x) = HookModel.hookRepresentation a h H g (K x))
(line : S → L) (ρ : Representation ℂ (lineGroup line) V) (hρ : IsUnitary ρ)
(I : Fin u → V →ₗᵢ[ℂ] hilbertSpace a)
(hI : ∀ i, Intertwines (V := V) (W := hilbertSpace a) ρ ((relabelledUnitary a e).comp (lineGroup line).subtype) (I i).toLinearMap)
include hK hρ hI in
omit [Fintype S] [DecidableEq S] [Fintype L] [DecidableEq L] in
lemma hook_copy_projection_intertwines :
Intertwines (Signed.lineRepresentation (odd := HookModel.IsOdd (h := h))
(k := HookModel.oddCount a h H) (line ∘ e.symm))
(Signed.lineRepresentation (odd := HookModel.IsOdd (h := h))
(k := HookModel.oddCount a h H) (line ∘ e.symm))
(copyProjection (fun i => K.comp (I i))).toLinearMap := by
let τ := hookLineTarget line a e h H
have hKI : ∀ i, Intertwines ρ τ (K.comp (I i)).toLinearMap := by
intro i g x
change K (I i (ρ g x)) = HookModel.hookRepresentation a h H (e.permCongr g.1) (K (I i x))
have hi : I i (ρ g x)=unitaryRepresentation a (e.permCongr g.1) (I i x) := hI i g x
rw [hi]
exact hK (e.permCongr g.1) _
have hp := copyProjection_intertwines ρ τ hρ
(fun g => Signed.representation_unitary (e.permCongr g.1)) (fun i => K.comp (I i)) hKI
intro g x
have hh := hp ((lineGroupCongr e line).symm g) x
change copyProjection (fun i => K.comp (I i))
(Signed.lineRepresentation (odd := HookModel.IsOdd (h := h)) (k := HookModel.oddCount a h H)
(line ∘ e.symm) ((lineGroupCongr e line) ((lineGroupCongr e line).symm g)) x) =
Signed.lineRepresentation (odd := HookModel.IsOdd (h := h)) (k := HookModel.oddCount a h H)
(line ∘ e.symm) ((lineGroupCongr e line) ((lineGroupCongr e line).symm g))
(copyProjection (fun i => K.comp (I i)) x) at hh
rw [MulEquiv.apply_symm_apply] at hh
exact hh
end LineCopy
section Filter
variable {G V E X : Type*} [Group G]
[NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
[NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
[NormedAddCommGroup X] [InnerProductSpace ℂ X] [FiniteDimensional ℂ X]
omit [FiniteDimensional ℂ X] in
lemma isotypic_filter_copy_projection (ρ : Representation ℂ G X) (τ : Representation ℂ G E)
(K : X →ₗᵢ[ℂ] E) (hK : ∀ g x, K (ρ g x)=τ g (K x))
(Q : E →L[ℂ] E) (hQ : ∀ x ∈ Isotypic.carrier ρ τ, Q x=x)
{u : ℕ} (I : Fin u → V →ₗᵢ[ℂ] X) :
Q * copyProjection (fun i => K.comp (I i)) = copyProjection (fun i => K.comp (I i)) := by
let f : Representation.IntertwiningMap ρ τ :=
{ toLinearMap := K.toLinearMap
isIntertwining' g := by ext x; exact hK g x }
have hk (x : X) : Q (K x)=K x := hQ _ (Isotypic.mem_carrier ρ τ f x)
ext x
simp only [mul_apply_eq_comp, copyProjection, sum_apply,
map_sum, ContinuousLinearMap.comp_apply, LinearIsometry.coe_toContinuousLinearMap,
LinearIsometry.coe_comp, Function.comp_apply, hk]
end Filter
end RowColumn
end
end
end OAI
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