| import Mathlib |
| import OAI.RepresentationTheory.PartialPermutation.SmallKernels |
|
|
| namespace OAI |
|
|
| section |
| open scoped Classical |
| open scoped BigOperators ComplexConjugate MonoidAlgebra |
| open scoped BigOperators ComplexConjugate |
| open scoped MonoidAlgebra BigOperators |
| open scoped BigOperators MonoidAlgebra Classical |
|
|
| attribute [local instance] Classical.propDecidable |
| namespace PartialPermutation |
| noncomputable section |
| variable {G : Type*} [Group G] [Fintype G] |
|
|
| def smallDegreeSum (G : Type*) [Group G] [Fintype G] (R : ℝ) : ℝ := |
| ∑ c : IrreducibleIndex G, if (irreducibleDegree c : ℝ) ≤ R then |
| (irreducibleDegree c : ℝ)^2 else 0 |
|
|
| lemma smallTypeKernel_energy (R : ℝ) : |
| (Fintype.card G : ℝ) * ∑ g, ‖smallTypeKernel G R g‖^2 = smallDegreeSum G R := by |
| rw [finite_group_plancherel] |
| unfold smallDegreeSum |
| apply Finset.sum_congr rfl |
| intro c _ |
| rw [complexFourier_smallTypeKernel] |
| split_ifs <;> simp [hsNormSq, LinearMap.trace_one, pow_two] |
|
|
| lemma kernel_hs_energy {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] |
| [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) |
| [Representation.IsIrreducible ρ] (H : Subgroup G) (k : H → ℂ) |
| (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : RightCosetCap f H B) : |
| (Module.finrank ℂ V : ℝ) * |
| hsNormSq (complexFourier (ρ.comp H.subtype) k * complexFourier ρ (fun g => f g)) ≤ |
| B * mass f * ((Fintype.card H : ℝ) * ∑ h, ‖k h‖^2) := by |
| have hc : (H.index : ℝ) * (Fintype.card H : ℝ) = (Fintype.card G : ℝ) := by |
| exact_mod_cast (by simpa only [Nat.card_eq_fintype_card] using H.index_mul_card) |
| have hi : (H.index : ℝ) ≠ 0 := by exact_mod_cast H.index_ne_zero_of_finite |
| calc |
| _ ≤ (Fintype.card G : ℝ) * ∑ g, ‖leftKernel H k f g‖^2 := by |
| rw [← complexFourier_leftKernel] |
| exact fourier_bessel ρ hρ _ |
| _ ≤ (Fintype.card G : ℝ) * (B / H.index * mass f * ∑ h, ‖k h‖^2) := |
| mul_le_mul_of_nonneg_left (leftKernel_l2 H k f B hf hcap) (by positivity) |
| _ = _ := by rw [← hc]; field_simp |
|
|
| lemma small_type_hs_bound {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] |
| [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) |
| [Representation.IsIrreducible ρ] (H : Subgroup G) (R : ℝ) |
| (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : LeftCosetCap f H B) : |
| (Module.finrank ℂ V : ℝ) * |
| hsNormSq (complexFourier ρ (fun g => f g) * |
| complexFourier (ρ.comp H.subtype) (smallTypeKernel H R)) ≤ |
| B * mass f * smallDegreeSum H R := by |
| have hi := kernel_hs_energy ρ hρ H (smallTypeKernel H R) (fun g => f g⁻¹) B |
| (fun g => hf _) (inverse_rightCap H f B hcap) |
| rw [mass_inverse, smallTypeKernel_energy] at hi |
| have hP := complexFourier_selfAdjoint (G := H) (ρ.comp H.subtype) |
| (fun (g : H) => hρ g) (smallTypeKernel H R) (smallTypeKernel_inv R) |
| have hA : LinearMap.adjoint (complexFourier ρ (fun g => (f g⁻¹ : ℂ))) = |
| complexFourier ρ (fun g => (f g : ℂ)) := by |
| rw [complexFourier_adjoint ρ hρ] |
| simp |
| rw [← hsNormSq_adjoint (_ * _), adjoint_mul_eq, hA, hP] at hi |
| exact hi |
|
|
| section Cover |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] |
| [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) |
| {ι : Type*} [Fintype ι] (P : ι → Module.End ℂ V) |
| (hP : ∀ i, LinearMap.adjoint (P i) = P i) |
| (hcover : ∀ c : isotypicComponents ℂ[G] ρ.asModule, |
| ∃ i, ∀ x ∈ componentSpace ρ c, P i x = x) |
|
|
| include ρ hρ hP hcover |
|
|
| lemma component_projection_cover_norm (v : V) : ‖v‖^2 ≤ ∑ i, ‖P i v‖^2 := by |
| let : Fintype (isotypicComponents ℂ[G] ρ.asModule) := Fintype.ofFinite _ |
| calc |
| _ = ∑ c, ‖(componentSpace ρ c).starProjection v‖^2 := |
| (sum_component_norm_sq ρ hρ v).symm |
| _ ≤ ∑ c, ∑ i, ‖(componentSpace ρ c).starProjection (P i v)‖^2 := by |
| apply Finset.sum_le_sum |
| intro c _ |
| obtain ⟨i, hi⟩ := hcover c |
| have he := congrArg (fun A : Module.End ℂ V => A v) |
| (projection_kernel_mul (componentSpace ρ c) (P i) (hP i) hi) |
| change (componentSpace ρ c).starProjection (P i v) = |
| (componentSpace ρ c).starProjection v at he |
| rw [← he] |
| exact Finset.single_le_sum (fun j _ => |
| sq_nonneg (‖(componentSpace ρ c).starProjection (P j v)‖)) (Finset.mem_univ i) |
| _ = _ := by |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact sum_component_norm_sq ρ hρ (P i v) |
|
|
| lemma component_projection_cover_hs (A : Module.End ℂ V) : |
| hsNormSq A ≤ ∑ i, hsNormSq (A * P i) := by |
| let e := stdOrthonormalBasis ℂ V |
| calc |
| _ = ∑ j, ‖LinearMap.adjoint A (e j)‖^2 := by |
| rw [← hsNormSq_adjoint A, hsNormSq_eq_sum e] |
| _ ≤ ∑ j, ∑ i, ‖P i (LinearMap.adjoint A (e j))‖^2 := |
| Finset.sum_le_sum (fun j _ => component_projection_cover_norm ρ hρ P hP hcover _) |
| _ = ∑ i, hsNormSq (P i * LinearMap.adjoint A) := by |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [hsNormSq_eq_sum e] |
| rfl |
| _ = _ := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [← hsNormSq_adjoint (A * P i), adjoint_mul_eq, hP] |
|
|
| end Cover |
| end |
| end PartialPermutation |
|
|
| end |
|
|
| end OAI |
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|
