| import Mathlib |
| import OAI.RepresentationTheory.PartialPermutation.ComponentGeometry |
| import OAI.RepresentationTheory.PartialPermutation.MatrixDegrees |
| import OAI.RepresentationTheory.PartialPermutation.Restriction |
| import OAI.RepresentationTheory.PartialPermutation.IsotypicKernels |
| import OAI.RepresentationTheory.PartialPermutation.DegreeEstimates |
| import OAI.RepresentationTheory.PartialPermutation.BlockSubgroups |
|
|
| 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 |
| open scoped MonoidAlgebra BigOperators |
| open scoped BigOperators |
|
|
| namespace PartialPermutation |
| open scoped MonoidAlgebra BigOperators ComplexConjugate |
|
|
| noncomputable section |
|
|
| lemma isotypic_joint_kernels {G M : Type*} [Group G] [Fintype G] |
| [NormedAddCommGroup M] [InnerProductSpace ℂ M] |
| [Module ℂ[G] M] [IsScalarTower ℂ ℂ[G] M] |
| [FiniteDimensional ℂ M] [Nontrivial M] |
| {b : ℕ} (H : Fin b → Subgroup G) (hgen : ⨆ i, H i = ⊤) |
| (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k) |
| (h : IsIsotypic ℂ[G] M) |
| (hu : ∀ (g : G) (x y : M), inner ℂ ((MonoidAlgebra.single g (1 : ℂ)) • x) |
| ((MonoidAlgebra.single g (1 : ℂ)) • y) = inner ℂ x y) : |
| ∃ d : Fin b → ℕ, (∀ i, 0 < d i) ∧ (∏ i, d i) ≤ Module.finrank ℂ M ∧ |
| ∀ i, ∃ k : H i → ℂ, |
| (∑ g, ‖k g‖^2 = (d i : ℝ)^2 / Fintype.card (H i)) ∧ |
| (∀ g, k g⁻¹ = conj (k g)) ∧ |
| (∀ x : M, ∑ g, k g • (MonoidAlgebra.single (g : G) (1 : ℂ)) • x = x) := by |
| classical |
| let (i : Fin b) : Module ℂ[H i] M := restrictGroupModule (H i) |
| let (i : Fin b) : IsScalarTower ℂ ℂ[H i] M := restrictGroupModule_tower (H i) |
| have hmodels (i : Fin b) : Nonempty (IsotypicKernelModel (H i) M) := by |
| apply exists_isotypicKernelModel (group_isotypic_restrict H hgen hcomm h i) |
| intro g x y |
| change inner ℂ ((MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype |
| (MonoidAlgebra.single g (1 : ℂ))) • x) |
| ((MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype |
| (MonoidAlgebra.single g (1 : ℂ))) • y) = _ |
| simpa [MonoidAlgebra.mapDomainAlgHom] using hu g x y |
| let F (i : Fin b) := Classical.choice (hmodels i) |
| have hAcomm (i j : Fin b) (hij : i ≠ j) (X Y) : Commute ((F i).lift X) ((F j).lift Y) := by |
| obtain ⟨r, hr⟩ := (F i).lift_image X |
| obtain ⟨s, hs⟩ := (F j).lift_image Y |
| rw [hr, hs] |
| have he (i : Fin b) (r : ℂ[H i]) : Algebra.lsmul ℂ ℂ M r = |
| Algebra.lsmul ℂ ℂ M (MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype r) := by |
| ext x |
| rfl |
| rw [he i r, he j s] |
| exact (subgroup_algebra_commute (H i) (H j) (hcomm i j hij) r s).map |
| (Algebra.lsmul ℂ ℂ M : ℂ[G] →ₐ[ℂ] Module.End ℂ M) |
| refine ⟨fun i => (F i).degree, fun i => (F i).degree_pos, ?_, ?_⟩ |
| · exact commuting_degrees_le_finrank (fun i => (F i).degree) |
| (fun i => (F i).degree_pos) (fun i => (F i).lift) hAcomm Finset.univ |
| · intro i |
| refine ⟨(F i).kernel, (F i).kernel_norm, (F i).kernel_inv, ?_⟩ |
| intro x |
| have ht := (F i).kernel_action x |
| change ∑ g, (F i).kernel g • (MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype |
| (MonoidAlgebra.single g (1 : ℂ))) • x = x at ht |
| simpa [MonoidAlgebra.mapDomainAlgHom] using ht |
|
|
| section ComponentKernel |
| variable {G V : Type*} [Group G] [Fintype G] |
| [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] |
| (ρ : Representation ℂ G V) {b : ℕ} (H : Fin b → Subgroup G) |
|
|
| lemma component_kernel (hb : 0 < b) (hD : 0 < Module.finrank ℂ V) (hρ : IsUnitary ρ) |
| (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k) |
| (c : isotypicComponents ℂ[↥(⨆ i, H i)] (Representation.asModule (ρ.comp (⨆ i, H i).subtype))) : |
| ∃ i, ∃ d : ℕ, ∃ k : H i → ℂ, |
| (d : ℝ) ≤ (Module.finrank ℂ V : ℝ) ^ (1 / (b : ℝ)) ∧ |
| (∑ g, ‖k g‖^2 = (d : ℝ)^2 / Fintype.card (H i)) ∧ |
| (∀ g, k g⁻¹ = conj (k g)) ∧ |
| (∀ x ∈ componentSpace (ρ.comp (⨆ i, H i).subtype) c, |
| complexFourier (ρ.comp (H i).subtype) k x = x) := by |
| classical |
| let J := ⨆ i, H i |
| let ρJ : Representation ℂ J V := ρ.comp J.subtype |
| let : NormedAddCommGroup c.val := |
| inferInstanceAs (NormedAddCommGroup (componentSpace ρJ c)) |
| let : InnerProductSpace ℂ c.val := |
| inferInstanceAs (InnerProductSpace ℂ (componentSpace ρJ c)) |
| let : FiniteDimensional ℂ c.val := |
| inferInstanceAs (FiniteDimensional ℂ (componentSpace ρJ c)) |
| let inc : c.val →ₗ[ℂ] V := c.val.subtype.restrictScalars ℂ |
| have hinc (g : J) (x : c.val) : |
| inc ((MonoidAlgebra.single g (1 : ℂ)) • x) = ρ (g : G) (inc x) := by |
| change ρJ.asAlgebraHom (MonoidAlgebra.single g 1) (inc x) = _ |
| rw [Representation.asAlgebraHom_single, one_smul] |
| rfl |
| have hu : ∀ (g : J) (x y : c.val), |
| inner ℂ ((MonoidAlgebra.single g (1 : ℂ)) • x) |
| ((MonoidAlgebra.single g (1 : ℂ)) • y) = inner ℂ x y := by |
| intro g x y |
| change inner ℂ (inc ((MonoidAlgebra.single g (1 : ℂ)) • x)) |
| (inc ((MonoidAlgebra.single g (1 : ℂ)) • y)) = inner ℂ (inc x) (inc y) |
| rw [hinc, hinc] |
| exact hρ (g : G) (inc x) (inc y) |
| obtain ⟨d, hdpos, hprod, hkernel⟩ := isotypic_joint_kernels |
| (M := c.val) (fun i => (H i).subgroupOf J) |
| (jointFactors_generate H) (jointFactors_commute H hcomm) |
| (IsIsotypic.isotypicComponents c.property) hu |
| have hdim : Module.finrank ℂ c.val ≤ Module.finrank ℂ V := |
| Submodule.finrank_le (componentSpace ρJ c) |
| obtain ⟨i, hi⟩ := exists_degree_le_rpow hb hD d (hprod.trans hdim) |
| obtain ⟨k, hk, hki, hka⟩ := hkernel i |
| let e := Subgroup.subgroupOfEquivOfLe (le_iSup H i) |
| let k' : H i → ℂ := fun g => k (e.symm g) |
| refine ⟨i, d i, k', hi, ?_, ?_, ?_⟩ |
| · have he : ∑ g, ‖k' g‖^2 = ∑ g, ‖k g‖^2 := |
| Fintype.sum_equiv e.symm.toEquiv _ _ (fun g => rfl) |
| rw [he, hk, Fintype.card_congr e.toEquiv] |
| · intro g |
| simp only [k', map_inv, hki] |
| · intro x hx |
| have ht := congrArg inc (hka ⟨x, hx⟩) |
| have hix : inc ⟨x, hx⟩ = x := rfl |
| have ht' : ∑ g : (H i).subgroupOf J, k g • ρ (g.val.val) x = x := by |
| simpa only [map_sum, map_smul, hinc, hix] using ht |
| rw [complexFourier, LinearMap.sum_apply] |
| calc |
| _ = ∑ g : (H i).subgroupOf J, k g • ρ (g.val.val) x := by |
| exact Fintype.sum_equiv e.symm.toEquiv _ _ (fun g => rfl) |
| _ = x := ht' |
|
|
| end ComponentKernel |
| end |
| end PartialPermutation |
|
|
| end |
|
|
| end OAI |
|
|
