| import Mathlib |
| import OAI.RepresentationTheory.PartialPermutation.Plancherel |
| import OAI.RepresentationTheory.PartialPermutation.ConvolutionKernels |
| import OAI.RepresentationTheory.PartialPermutation.IsotypicProjections |
|
|
| 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] |
|
|
| lemma centralCharacterKernel_norm_sq {W : Type*} [NormedAddCommGroup W] |
| [InnerProductSpace ℂ W] [FiniteDimensional ℂ W] |
| (τ : Representation ℂ G W) [Representation.IsIrreducible τ] (hτ : IsUnitary τ) : |
| ∑ g, ‖centralCharacterKernel τ g‖^2 = |
| (Module.finrank ℂ W : ℝ)^2 / Fintype.card G := by |
| have hN : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero |
| simp only [centralCharacterKernel, unitary_character_inv τ hτ, norm_mul, |
| norm_div, Complex.norm_natCast, Complex.norm_conj, mul_pow, |
| ← Finset.mul_sum, character_norm_sq_sum τ hτ] |
| field_simp |
|
|
| lemma centralCharacterKernel_inv {W : Type*} [NormedAddCommGroup W] |
| [InnerProductSpace ℂ W] [FiniteDimensional ℂ W] |
| (τ : Representation ℂ G W) (hτ : IsUnitary τ) (g : G) : |
| centralCharacterKernel τ g⁻¹ = conj (centralCharacterKernel τ g) := by |
| simp [centralCharacterKernel, unitary_character_inv τ hτ] |
|
|
| lemma kernel_hs_global (H : Subgroup G) (k : H → ℂ) |
| (d : ℕ) (hk : ∑ h : H, ‖k h‖^2 = (d : ℝ)^2 / Fintype.card H) |
| (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : RightCosetCap f H B) : |
| ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ) * |
| hsNormSq (V := irreducibleSpace c) |
| (complexFourier ((irreducibleRep c).comp H.subtype) k * |
| complexFourier (irreducibleRep c) (fun g => (f g : ℂ))) ≤ |
| B * mass f * (d : ℝ)^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 |
| have hh : (Fintype.card H : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero |
| calc |
| _ = (Fintype.card G : ℝ) * ∑ g, ‖leftKernel H k f g‖^2 := by |
| rw [finite_group_plancherel] |
| apply Finset.sum_congr rfl |
| intro c _ |
| rw [complexFourier_leftKernel] |
| _ ≤ (Fintype.card G : ℝ) * (B / H.index * mass f * ∑ h : H, ‖k h‖^2) := |
| mul_le_mul_of_nonneg_left (leftKernel_l2 H k f B hf hcap) (by positivity) |
| _ = _ := by rw [hk, ← hc]; field_simp |
|
|
| lemma kernel_hs_global_right (H : Subgroup G) (k : H → ℂ) |
| (d : ℕ) (hk : ∑ h : H, ‖k h‖^2 = (d : ℝ)^2 / Fintype.card H) |
| (hki : ∀ h, k h⁻¹ = conj (k h)) |
| (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : LeftCosetCap f H B) : |
| ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ) * |
| hsNormSq (V := irreducibleSpace c) |
| (complexFourier (irreducibleRep c) (fun g => (f g : ℂ)) * |
| complexFourier ((irreducibleRep c).comp H.subtype) k) ≤ |
| B * mass f * (d : ℝ)^2 := by |
| have hl := kernel_hs_global H k d hk (fun g => f g⁻¹) B |
| (fun g => hf _) (inverse_rightCap H f B hcap) |
| rw [mass_inverse] at hl |
| convert hl using 1 |
| apply Finset.sum_congr rfl |
| intro c _ |
| have hP := complexFourier_selfAdjoint (G := H) (V := irreducibleSpace c) |
| ((irreducibleRep c).comp H.subtype) (fun (g : H) => irreducibleRep_unitary c g) k hki |
| have hA : LinearMap.adjoint (E := irreducibleSpace c) (F := irreducibleSpace c) |
| (complexFourier (irreducibleRep c) (fun g => (f g⁻¹ : ℂ))) = |
| complexFourier (irreducibleRep c) (fun g => (f g : ℂ)) := by |
| rw [complexFourier_adjoint _ (irreducibleRep_unitary c)] |
| simp |
| rw [← hsNormSq_adjoint (V := irreducibleSpace c) |
| (complexFourier ((irreducibleRep c).comp H.subtype) k * |
| complexFourier (irreducibleRep c) (fun g => (f g⁻¹ : ℂ))), adjoint_mul_eq, hP, hA] |
|
|
| def fullIsotypicProjection {V W : Type*} [AddCommGroup V] [Module ℂ V] |
| [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W] |
| (ρ : Representation ℂ G V) (H : Subgroup G) (τ : Representation ℂ H W) : |
| Module.End ℂ V := complexFourier (ρ.comp H.subtype) (centralCharacterKernel τ) |
|
|
| lemma centralCharacterKernel_equiv {V W : Type*} [AddCommGroup V] [Module ℂ V] |
| [FiniteDimensional ℂ V] [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W] |
| {ρ : Representation ℂ G V} {σ : Representation ℂ G W} |
| (e : Representation.Equiv ρ σ) : centralCharacterKernel ρ = centralCharacterKernel σ := by |
| funext g |
| simp only [centralCharacterKernel, Representation.char_iso e, e.toLinearEquiv.finrank_eq] |
|
|
| lemma fullIsotypicProjection_eq_component {V W : Type*} [NormedAddCommGroup V] |
| [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] |
| [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W] |
| (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (H : Subgroup G) |
| (τ : Representation ℂ H W) |
| (c : isotypicComponents ℂ[H] (Representation.asModule (ρ.comp H.subtype))) |
| (e : Representation.Equiv τ (componentRep (ρ.comp H.subtype) c)) : |
| fullIsotypicProjection ρ H τ = |
| (componentSpace (ρ.comp H.subtype) c).starProjection.toLinearMap := by |
| rw [fullIsotypicProjection, centralCharacterKernel_equiv e] |
| exact component_kernel_eq_projection (G := H) (ρ.comp H.subtype) |
| (fun (g : H) => hρ g) c |
|
|
| theorem central_isotypic_global (H : Subgroup G) {W : Type*} [NormedAddCommGroup W] |
| [InnerProductSpace ℂ W] [FiniteDimensional ℂ W] |
| (τ : Representation ℂ H W) [Representation.IsIrreducible τ] (hτ : IsUnitary τ) |
| (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : LeftCosetCap f H B) : |
| ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ) * |
| hsNormSq (V := irreducibleSpace c) |
| (complexFourier (irreducibleRep c) (fun g => (f g : ℂ)) * |
| fullIsotypicProjection (irreducibleRep c) H τ) ≤ |
| B * mass f * (Module.finrank ℂ W : ℝ)^2 := |
| kernel_hs_global_right H (centralCharacterKernel τ) _ |
| (centralCharacterKernel_norm_sq τ hτ) (centralCharacterKernel_inv τ hτ) f B hf hcap |
|
|
| end |
| end PartialPermutation |
|
|
| end |
|
|
| end OAI |
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|
