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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.ProjectionEnergy
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
namespace PartialPermutation
noncomputable section
variable {G : Type*} [Group G] [Fintype G]
def smallTypeKernel (G : Type*) [Group G] [Fintype G] (R : ℝ) : G → ℂ :=
fun g => ∑ c : IrreducibleIndex G,
if (irreducibleDegree c : ℝ) ≤ R then centralCharacterKernel (irreducibleRep c) g else 0
lemma smallTypeKernel_inv (R : ℝ) (g : G) :
smallTypeKernel G R g⁻¹ = conj (smallTypeKernel G R g) := by
simp only [smallTypeKernel, map_sum]
apply Finset.sum_congr rfl
intro c _
split_ifs <;> simp [centralCharacterKernel_inv _ (irreducibleRep_unitary c)]
lemma complexFourier_smallTypeKernel {V : Type*} [AddCommGroup V] [Module ℂ V]
[FiniteDimensional ℂ V] (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
(R : ℝ) : complexFourier ρ (smallTypeKernel G R) =
if (Module.finrank ℂ V : ℝ) ≤ R then 1 else 0 := by
classical
obtain ⟨c, ⟨e⟩, hc⟩ := irreducibleRep_complete ρ
have hdeg : irreducibleDegree c = Module.finrank ℂ V := e.toLinearEquiv.finrank_eq
have he (d : IrreducibleIndex G) :
Nonempty (Representation.Equiv (irreducibleRep d) ρ) ↔ d = c := by
exact ⟨hc d, fun h => h ▸ ⟨e⟩⟩
unfold smallTypeKernel
rw [complexFourier_sum]
calc
_ = ∑ d : IrreducibleIndex G, if d = c then
(if (Module.finrank ℂ V : ℝ) ≤ R then (1 : Module.End ℂ V) else 0) else 0 := by
apply Finset.sum_congr rfl
intro d _
by_cases hd : (irreducibleDegree d : ℝ) ≤ R
· simp only [hd, ite_true, centralCharacterKernel_fourier]
by_cases hdc : d = c
· subst d
rw [hdeg] at hd
simp [he, hd]
· simp [he, hdc]
· simp only [hd, ite_false, complexFourier]
by_cases hdc : d = c
· subst d
simp [hdeg] at hd
simp [hd]
· simp [hdc]
_ = _ := by simp
@[simp] lemma hsNormSq_one {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
[FiniteDimensional ℂ V] : hsNormSq (1 : Module.End ℂ V) = Module.finrank ℂ V := by
simp [hsNormSq, LinearMap.trace_one]
lemma degree_le_of_kernel_identity {V : Type*} [NormedAddCommGroup V]
[InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
(ρ : Representation ℂ G V) [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
(k : G → ℂ) (d : ℕ) (hk : ∑ g, ‖k g‖^2 = (d : ℝ)^2 / Fintype.card G)
(hact : complexFourier ρ k = 1) : Module.finrank ℂ V ≤ d := by
have h := fourier_bessel ρ hρ k
have hN : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
rw [hact, hsNormSq_one, hk] at h
have hd : (Module.finrank ℂ V : ℝ)^2 ≤ (d : ℝ)^2 := by
convert h using 1 <;> field_simp
exact_mod_cast (sq_le_sq₀ (by positivity) (by positivity)).mp hd
lemma smallTypeKernel_identity_of_kernel {V : Type*} [NormedAddCommGroup V]
[InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
(ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
(k : G → ℂ) (d : ℕ) (R : ℝ) (hd : (d : ℝ) ≤ R)
(hk : ∑ g, ‖k g‖^2 = (d : ℝ)^2 / Fintype.card G)
(hact : complexFourier ρ k = 1) : complexFourier ρ (smallTypeKernel G R) = 1 := by
apply linearMap_eq_on_simples ρ
intro σ hσ x
have := hσ
have hu : IsUnitary σ.toRepresentation := fun g x y => hρ g x y
have hs : complexFourier σ.toRepresentation k = 1 := by
ext y
rw [complexFourier_subrep, hact]
rfl
have hdeg := degree_le_of_kernel_identity σ.toRepresentation hu k d hk hs
have hdegR : (Module.finrank ℂ σ.toSubmodule : ℝ) ≤ R := (Nat.cast_le.mpr hdeg).trans hd
rw [← complexFourier_subrep, complexFourier_smallTypeKernel, ite_eq_left hdegR]
rfl
end
end PartialPermutation
end
end OAI
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