| import Mathlib |
| import OAI.Probability.BinarySweep.Representations.MultiplicityBound |
| import OAI.Probability.BinarySweep.MatrixBounds.PositiveTraceRestriction |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| open scoped BigOperators Classical ComplexOrder |
|
|
| namespace BinaryCoordinateSweeps.Irrep |
| open Representation |
|
|
| variable {G V W : Type*} [Group G] [Fintype G] |
| [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] |
| [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W] |
| (ρ : Representation ℂ G V) (σ : Representation ℂ G W) |
|
|
| omit [Fintype G] [FiniteDimensional ℂ W] in |
| lemma commuting_eigenspace_map (T : Module.End ℂ W) |
| (hT : ∀ g w, T (σ g w) = σ g (T w)) (μ : ℂ) (g : G) : |
| (T.eigenspace μ).map (σ g) = T.eigenspace μ := by |
| apply le_antisymm |
| · rintro w ⟨v,hv,rfl⟩ |
| apply Module.End.mem_eigenspace_iff.mpr |
| rw [hT,Module.End.mem_eigenspace_iff.mp hv,map_smul] |
| · intro w hw |
| refine ⟨σ g⁻¹ w, ?_, ?_⟩ |
| · apply Module.End.mem_eigenspace_iff.mpr |
| rw [hT,Module.End.mem_eigenspace_iff.mp hw,map_smul] |
| · rw [← Module.End.mul_apply,← map_mul,mul_inv_cancel,map_one,Module.End.one_apply] |
|
|
| def eigenspaceProjectionIntertwiner (T : Module.End ℂ W) |
| (hT : ∀ g w, T (σ g w) = σ g (T w)) |
| (hσ : ∀ g w, ‖σ g w‖ = ‖w‖) (μ : ℂ) : IntertwiningMap σ σ where |
| toLinearMap := (T.eigenspace μ).starProjection.toLinearMap |
| isIntertwining' g := by |
| ext w |
| change (T.eigenspace μ).starProjection (σ g w) = σ g ((T.eigenspace μ).starProjection w) |
| let U : W →ₗᵢ[ℂ] W := {toLinearMap := σ g, norm_map' := hσ g} |
| have he := U.map_starProjection (T.eigenspace μ) w |
| change σ g ((T.eigenspace μ).starProjection w) = |
| ((T.eigenspace μ).map (σ g)).starProjection (σ g w) at he |
| simpa only [commuting_eigenspace_map σ T hT μ g] using he.symm |
|
|
| omit [Fintype G] [FiniteDimensional ℂ V] in |
| lemma eigenvalue_multiplicity_ge [ρ.IsIrreducible] (T : Module.End ℂ W) |
| (hT : ∀ g w, T (σ g w) = σ g (T w)) |
| (hσ : ∀ g w, ‖σ g w‖ = ‖w‖) (μ : ℂ) |
| {w : W} (hw : w ∈ isotypicSpan ρ σ) (he : T w = μ • w) (hn : w ≠ 0) : |
| Module.finrank ℂ V ≤ Module.finrank ℂ (T.eigenspace μ) := by |
| let P := eigenspaceProjectionIntertwiner σ T hT hσ μ |
| have hP : ∃ f : IntertwiningMap ρ σ, P.comp f ≠ 0 := by |
| by_contra! hall |
| have hz : P w = 0 := by |
| change w ∈ ⨆ f : IntertwiningMap ρ σ, LinearMap.range f.toLinearMap at hw |
| refine Submodule.iSup_induction _ (motive := fun x => P x = 0) hw ?_ (map_zero P) ?_ |
| · intro f x hx |
| obtain ⟨v,rfl⟩ := hx |
| exact DFunLike.congr_fun (hall f) v |
| · intro x y hx hy |
| rw [map_add,hx,hy,zero_add] |
| have hpw : P w = w := Submodule.starProjection_eq_self_iff.mpr |
| (Module.End.mem_eigenspace_iff.mpr he) |
| exact hn (hpw.symm.trans hz) |
| obtain ⟨f,hf⟩ := hP |
| have hi := (IsIrreducible.injective_or_eq_zero (P.comp f)).resolve_right hf |
| let Q : V →ₗ[ℂ] T.eigenspace μ := (P.comp f).toLinearMap.codRestrict _ |
| (fun v => (T.eigenspace μ).starProjection_apply_mem (f v)) |
| apply LinearMap.finrank_le_finrank_of_injective (f:=Q) |
| intro v v' hv |
| exact hi (congrArg Subtype.val hv) |
|
|
| omit [Fintype G] [FiniteDimensional ℂ V] in |
| theorem isotypic_eigenvalue_trace_le [ρ.IsIrreducible] (T : Module.End ℂ W) |
| (hpos : T.IsPositive) (hT : ∀ g w, T (σ g w) = σ g (T w)) |
| (hσ : ∀ g w, ‖σ g w‖ = ‖w‖) {μ : ℝ} (hμ : 0 ≤ μ) |
| {w : W} (hw : w ∈ isotypicSpan ρ σ) (he : T w = (μ:ℂ) • w) (hn : w ≠ 0) : |
| (Module.finrank ℂ V : ℝ) * μ ≤ (LinearMap.trace ℂ W T).re := by |
| let E := T.eigenspace (μ:ℂ) |
| have hE : ∀ x ∈ E, T x ∈ E := by |
| intro x hx |
| rw [Module.End.mem_eigenspace_iff] at hx ⊢ |
| rw [hx,map_smul,hx] |
| have hr : T.restrict hE = (μ:ℂ) • (LinearMap.id : E →ₗ[ℂ] E) := by |
| ext x |
| exact Module.End.mem_eigenspace_iff.mp x.property |
| have ht := positive_trace_restrict_le E T hpos hE |
| rw [hr,map_smul,LinearMap.trace_id] at ht |
| have hd := eigenvalue_multiplicity_ge ρ σ T hT hσ (μ:ℂ) hw he hn |
| have hd' : (Module.finrank ℂ V : ℝ) ≤ (Module.finrank ℂ E : ℝ) := by exact_mod_cast hd |
| have ht' : μ * (Module.finrank ℂ E : ℝ) ≤ (LinearMap.trace ℂ W T).re := by |
| simpa using ht |
| nlinarith |
|
|
| end BinaryCoordinateSweeps.Irrep |
|
|
| end |
|
|
| open scoped BigOperators Classical ComplexOrder |
|
|
| namespace BinaryCoordinateSweeps.Irrep |
| open Representation |
|
|
| variable {G V W : Type*} [Group G] [Fintype G] |
| [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] |
| [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W] |
| (ρ : Representation ℂ G V) (σ : Representation ℂ G W) |
|
|
| def commutingIntertwiner (T : Module.End ℂ W) |
| (hT : ∀ g w, T (σ g w) = σ g (T w)) : IntertwiningMap σ σ where |
| toLinearMap := T |
| isIntertwining' g := by ext w; exact hT g w |
| |
| omit [Fintype G] [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in |
| lemma commuting_isotypic_invariant (T : Module.End ℂ W) |
| (hT : ∀ g w, T (σ g w) = σ g (T w)) : |
| ∀ w ∈ isotypicSpan ρ σ, T w ∈ isotypicSpan ρ σ := by |
| intro w hw |
| apply Submodule.iSup_induction (fun f : IntertwiningMap ρ σ => LinearMap.range f.toLinearMap) |
| (motive := fun w => T w ∈ isotypicSpan ρ σ) hw |
| · intro f w hw |
| obtain ⟨v,rfl⟩ := hw |
| let F := (commutingIntertwiner σ T hT).comp f |
| exact (le_iSup (fun f : IntertwiningMap ρ σ => LinearMap.range f.toLinearMap) F) ⟨v,rfl⟩ |
| · simp |
| · intro x y hx hy |
| simpa only [map_add] using Submodule.add_mem _ hx hy |
|
|
| omit [Fintype G] [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in |
| lemma isotypic_restrict_positive (T : Module.End ℂ W) (hpos : T.IsPositive) |
| (hT : ∀ g w, T (σ g w) = σ g (T w)) : |
| (T.restrict (commuting_isotypic_invariant ρ σ T hT)).IsPositive := |
| ⟨hpos.isSymmetric.restrict_invariant _, fun x => hpos.2 x.val⟩ |
|
|
| omit [Fintype G] [FiniteDimensional ℂ V] in |
| theorem isotypic_restriction_cap [ρ.IsIrreducible] (T : Module.End ℂ W) |
| (hpos : T.IsPositive) (hT : ∀ g w, T (σ g w) = σ g (T w)) |
| (hσ : ∀ g w, ‖σ g w‖ = ‖w‖) : |
| (((LinearMap.trace ℂ W T).re : ℂ) • (LinearMap.id : isotypicSpan ρ σ →ₗ[ℂ] isotypicSpan ρ σ) - |
| (Module.finrank ℂ V : ℂ) • T.restrict (commuting_isotypic_invariant ρ σ T hT)).IsPositive := by |
| let S := isotypicSpan ρ σ |
| let R := T.restrict (commuting_isotypic_invariant ρ σ T hT) |
| have hR : R.IsPositive := isotypic_restrict_positive ρ σ T hpos hT |
| let b := hR.isSymmetric.eigenvectorBasis rfl |
| apply (LinearMap.posSemidef_toMatrix_iff b).mp |
| rw [map_sub,map_smul,map_smul,LinearMap.toMatrix_id] |
| rw [show LinearMap.toMatrix b.toBasis b.toBasis R = |
| Matrix.diagonal (fun i => (hR.isSymmetric.eigenvalues rfl i : ℂ)) from |
| hR.isSymmetric.toMatrix_eigenvectorBasis rfl] |
| rw [← Matrix.diagonal_one,← Matrix.diagonal_smul,← Matrix.diagonal_smul,Matrix.diagonal_sub] |
| apply Matrix.posSemidef_diagonal_iff.mpr |
| intro i |
| have hi := isotypic_eigenvalue_trace_le ρ σ T hpos hT hσ |
| (hR.nonneg_eigenvalues rfl i) (b i).property |
| (congrArg Subtype.val (hR.isSymmetric.apply_eigenvectorBasis rfl i)) |
| (fun hz => (hR.isSymmetric.hasEigenvector_eigenvectorBasis rfl i).2 (Subtype.ext hz)) |
| change 0 ≤ ((LinearMap.trace ℂ W T).re : ℂ)*1 - _ |
| have he : (0:ℂ) ≤ (((LinearMap.trace ℂ W T).re - |
| (Module.finrank ℂ V : ℝ)*hR.isSymmetric.eigenvalues rfl i : ℝ):ℂ) := by |
| exact_mod_cast sub_nonneg.mpr hi |
| simpa using he |
|
|
| end BinaryCoordinateSweeps.Irrep |
|
|
| end |
|
|
| end OAI |
|
|
