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c8720e4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | import Mathlib
import OAI.RepresentationTheory.PartialPermutation.Components
import OAI.RepresentationTheory.PartialPermutation.CentralKernels
namespace OAI
section
open scoped Classical
open scoped BigOperators ComplexConjugate MonoidAlgebra
open scoped BigOperators ComplexConjugate
open scoped MonoidAlgebra BigOperators
namespace PartialPermutation
noncomputable section
open scoped MonoidAlgebra BigOperators
section CompleteKernels
variable {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
[FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
def componentRep (c : isotypicComponents ℂ[G] ρ.asModule) :=
(Subrepresentation.ofSubmodule' (componentSimple ρ c)).toRepresentation
instance componentRep_irreducible (c : isotypicComponents ℂ[G] ρ.asModule) :
Representation.IsIrreducible (componentRep ρ c) :=
(subrep_irreducible_iff ρ _).mpr (componentSimple_simple ρ c)
lemma componentRep_equiv_iff (c d : isotypicComponents ℂ[G] ρ.asModule) :
Nonempty (Representation.Equiv (componentRep ρ c) (componentRep ρ d)) ↔ c = d := by
constructor
· rintro ⟨e⟩
apply componentCharacter_injective ρ
exact Subtype.ext (Representation.char_iso e)
· rintro rfl
exact ⟨Representation.Equiv.refl _⟩
omit [FiniteDimensional ℂ V] in
lemma complexFourier_subrep (σ : Subrepresentation ρ) (a : G → ℂ) (x : σ) :
((complexFourier σ.toRepresentation a x : σ) : V) = complexFourier ρ a x := by
simp only [complexFourier, LinearMap.sum_apply, LinearMap.smul_apply,
Submodule.coe_sum, Submodule.coe_smul]
rfl
omit [FiniteDimensional ℂ V] in
lemma complexFourier_sum {ι : Type*} [Fintype ι] (a : ι → G → ℂ) :
complexFourier ρ (fun g => ∑ i, a i g) = ∑ i, complexFourier ρ (a i) := by
simp only [complexFourier, Finset.sum_smul]
exact Finset.sum_comm
lemma sum_character_kernels_identity
[Fintype (isotypicComponents ℂ[G] ρ.asModule)] :
complexFourier ρ (fun g => ∑ c, centralCharacterKernel (componentRep ρ c) g) = 1 := by
classical
let K : G → ℂ := fun g => ∑ c, centralCharacterKernel (componentRep ρ c) g
have hs (σ : Subrepresentation ρ) [Representation.IsIrreducible σ.toRepresentation] :
complexFourier σ.toRepresentation K = 1 := by
let χ : OccurringTypes ρ := ⟨σ.toRepresentation.character, σ, inferInstance, rfl⟩
obtain ⟨c, hc⟩ := componentCharacter_surjective ρ χ
have he : Nonempty (Representation.Equiv (componentRep ρ c) σ.toRepresentation) := by
exact irreducible_equiv_of_character_eq σ.toRepresentation (componentRep ρ c)
(congrArg Subtype.val hc).symm
rw [complexFourier_sum]
rw [Finset.sum_eq_single c]
· rw [centralCharacterKernel_fourier, ite_eq_left he]
· intro d _ hdc
rw [centralCharacterKernel_fourier, ite_eq_right]
rintro ⟨e⟩
obtain ⟨ec⟩ := he
exact hdc ((componentRep_equiv_iff ρ d c).mp ⟨e.trans ec.symm⟩)
· simp
let E (S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S}) : Submodule ℂ V :=
S.val.restrictScalars ℂ
have htop : (⨆ S, E S) = ⊤ := by
change (⨆ S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S},
S.val.restrictScalars ℂ) = ⊤
rw [← Submodule.restrictScalars_iSup]
have ht : (⨆ S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S}, S.val) = ⊤ :=
(sSup_eq_iSup' {S : Submodule ℂ[G] ρ.asModule | IsSimpleModule ℂ[G] S}).symm.trans
(IsSemisimpleModule.sSup_simples_eq_top ℂ[G] ρ.asModule)
rw [ht, Submodule.restrictScalars_top]
have hker : (⊤ : Submodule ℂ V) ≤ LinearMap.ker (complexFourier ρ K - 1) := by
rw [← htop]
refine iSup_le fun S => ?_
intro x hx
let σ := Subrepresentation.ofSubmodule' S.val
have : Representation.IsIrreducible σ.toRepresentation :=
(subrep_irreducible_iff ρ σ).mpr S.property
have ht := congrArg (fun A : Module.End ℂ σ.toSubmodule => ((A ⟨x, hx⟩ : σ.toSubmodule) : V)) (hs σ)
rw [complexFourier_subrep] at ht
simpa using sub_eq_zero.mpr ht
ext x
have hx := hker (Submodule.mem_top : x ∈ (⊤ : Submodule ℂ V))
simpa [LinearMap.mem_ker, sub_eq_zero] using hx
end CompleteKernels
end
end PartialPermutation
end
end OAI
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