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namespace OAI
open scoped Classical BigOperators ComplexConjugate MonoidAlgebra
namespace PartialPermutation
noncomputable section
def IsUnitary {G V : Type*} [Group G] [NormedAddCommGroup V]
[InnerProductSpace ℂ V] (ρ : Representation ℂ G V) : Prop :=
∀ g x y, inner ℂ (ρ g x) (ρ g y) = inner ℂ x y
section
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
[FiniteDimensional ℂ V]
def hsNormSq (A : Module.End ℂ V) : ℝ :=
(LinearMap.trace ℂ V (LinearMap.adjoint A * A)).re
end
def complexFourier {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
(ρ : Representation ℂ G V) (f : G → ℂ) : Module.End ℂ V := ∑ g, f g • ρ g
section
variable {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
[FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
noncomputable def componentSimple (c : isotypicComponents ℂ[G] ρ.asModule) :
Submodule ℂ[G] ρ.asModule := c.property.choose
instance components_finite : Finite (isotypicComponents ℂ[G] ρ.asModule) := by
let : IsNoetherian ℂ[G] ρ.asModule := isNoetherian_of_tower ℂ inferInstance
infer_instance
end
section
variable (G : Type*) [Group G] [Fintype G]
def regularRep : Representation ℂ G (EuclideanSpace ℂ G) where
toFun g :=
{ toFun := fun f => WithLp.toLp 2 (fun h => f (g⁻¹ * h))
map_add' := by intros; rfl
map_smul' := by intros; rfl }
map_one' := by ext f h; simp
map_mul' := by
intro x y
ext f h
change f ((x * y)⁻¹ * h) = f (y⁻¹ * (x⁻¹ * h))
simp [mul_assoc]
end
section
variable {G : Type*} [Group G] [Fintype G]
abbrev IrreducibleIndex (G : Type*) [Group G] [Fintype G] :=
isotypicComponents ℂ[G] (regularRep G).asModule
noncomputable instance irreducibleIndex_fintype : Fintype (IrreducibleIndex G) :=
Fintype.ofFinite _
abbrev irreducibleSpace (c : IrreducibleIndex G) :=
(Subrepresentation.ofSubmodule' (componentSimple (regularRep G) c)).toSubmodule
abbrev irreducibleDegree (c : IrreducibleIndex G) := Module.finrank ℂ (irreducibleSpace c)
end
def mass {G : Type*} [Fintype G] (f : G → ℝ) : ℝ := ∑ g, f g
def LeftCosetCap {G : Type*} [Group G] [Fintype G]
(f : G → ℝ) (H : Subgroup G) (B : ℝ) : Prop :=
by classical exact ∀ g, ∑ h : H, f (g * h) ≤ B / H.index
def blockSubgroup {X : Type*} (M : Set X) : Subgroup (Equiv.Perm X) where
carrier := {g | ∀ x, x ∉ M → g x = x}
one_mem' := by simp
mul_mem' := by
intro g h hg hh x hx
simp only [Set.mem_ofPred_eq] at hg hh ⊢
simp [Equiv.Perm.mul_apply, hh x hx, hg x hx]
inv_mem' := by
intro g hg x hx
exact ((Equiv.eq_symm_apply g).mpr (hg x hx)).symm
def inverseDegreeSum (G : Type*) [Group G] [Fintype G] (u : ℝ) : ℝ :=
∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ)^(-u)
def symmetricInverseDegreeSum (n : ℕ) (u : ℝ) : ℝ :=
inverseDegreeSum (Equiv.Perm (Fin n)) u
def symmetricDegreeConstant (u : ℝ) : ℝ :=
sSup {x : ℝ | ∃ n : ℕ, 1 ≤ n ∧ x = symmetricInverseDegreeSum n u}
def ComplementaryCosetsFiniteStatement : Prop :=
∀ (X : Type*) [Fintype X] [DecidableEq X] (b : ℕ) (_ : 0 < b) (M : Fin b → Set X),
(∀ i, (M i).Nonempty) →
Pairwise (fun i j => Disjoint (M i) (M j)) → (⋃ i, M i) = Set.univ →
∀ (V : Type) [NormedAddCommGroup V] [InnerProductSpace ℂ V]
[FiniteDimensional ℂ V] (ρ : Representation ℂ (Equiv.Perm X) V),
Representation.IsIrreducible ρ → IsUnitary ρ →
∀ (f : Equiv.Perm X → ℝ) (B u : ℝ),
(∀ g, 0 ≤ f g) → mass f ≤ 1 → 0 < u →
(∀ i, LeftCosetCap f (blockSubgroup (M i)) B) →
(hsNormSq (complexFourier ρ (fun g => f g)) ≤
b * B * symmetricDegreeConstant u *
(Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b : ℝ))) ∧
(‖LinearMap.toContinuousLinearMap (complexFourier ρ (fun g => f g))‖^2 ≤
b * B * symmetricDegreeConstant u *
(Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b : ℝ)))
end
end PartialPermutation
end OAI
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