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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
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