| import Mathlib |
| import OAI.RepresentationTheory.PartialPermutation.DegreeTransfer |
|
|
| 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 |
| open scoped MonoidAlgebra BigOperators |
| open scoped BigOperators |
| |
| namespace PartialPermutation |
| noncomputable section |
|
|
| lemma inverseDegreeSum_nonneg (G : Type*) [Group G] [Fintype G] (u : ℝ) : |
| 0 ≤ inverseDegreeSum G u := by |
| exact Finset.sum_nonneg (fun c _ => Real.rpow_nonneg (by positivity) _) |
|
|
| lemma smallDegreeSum_le_inverseDegreeSum (G : Type*) [Group G] [Fintype G] |
| {u R : ℝ} (hu : 0 < u) (hR : 0 < R) : |
| smallDegreeSum G R ≤ R^(u+2) * inverseDegreeSum G u := by |
| rw [smallDegreeSum, inverseDegreeSum, Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro c _ |
| split_ifs with hc |
| · have hd : 0 < (irreducibleDegree c : ℝ) := by |
| have : Nontrivial (irreducibleSpace c) := |
| IsSimpleModule.nontrivial ℂ[G] (irreducibleRep c).asModule |
| exact_mod_cast (Module.finrank_pos : 0 < irreducibleDegree c) |
| calc |
| (irreducibleDegree c : ℝ)^2 = |
| (irreducibleDegree c : ℝ)^(u+2) * (irreducibleDegree c : ℝ)^(-u) := by |
| rw [← Real.rpow_add hd] |
| ring_nf |
| rw [Real.rpow_two] |
| _ ≤ _ := mul_le_mul_of_nonneg_right |
| (Real.rpow_le_rpow hd.le hc (by linarith)) (Real.rpow_nonneg hd.le _) |
| · positivity |
|
|
| variable {G V : Type*} [Group G] [Fintype G] |
| [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] |
| (ρ : Representation ℂ G V) {b : ℕ} (H : Fin b → Subgroup G) |
|
|
| theorem inverse_degree_sum_transfer [Representation.IsIrreducible ρ] |
| (hb : 0 < b) (hρ : IsUnitary ρ) |
| (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k) |
| (f : G → ℝ) (B u : ℝ) (hf : ∀ g, 0 ≤ f g) (hu : 0 < u) |
| (hcap : ∀ i, LeftCosetCap f (H i) B) : |
| hsNormSq (complexFourier ρ (fun g => f g)) ≤ |
| B * mass f * (Module.finrank ℂ V : ℝ)^(-1 + (u+2)/(b : ℝ)) * |
| ∑ i, inverseDegreeSum (H i) u := by |
| have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule |
| have hD : 0 < (Module.finrank ℂ V : ℝ) := by exact_mod_cast (Module.finrank_pos : 0 < Module.finrank ℂ V) |
| have hB : 0 ≤ B := by |
| let i : Fin b := ⟨0, hb⟩ |
| have hp : (0 : ℝ) < (H i).index := by |
| exact_mod_cast Nat.pos_of_ne_zero (H i).index_ne_zero_of_finite |
| have hz : 0 ≤ B / (H i).index := |
| (Finset.sum_nonneg (fun (g : H i) _ => hf (1*g))).trans (hcap i 1) |
| simpa only [div_mul_cancel₀ B hp.ne'] using mul_nonneg hz hp.le |
| have hm : 0 ≤ mass f := Finset.sum_nonneg (fun g _ => hf g) |
| have ht := small_degree_sum_transfer ρ H hb hρ hcomm f B hf hcap |
| have hs : ∑ i, smallDegreeSum (H i) ((Module.finrank ℂ V : ℝ)^(1 / (b : ℝ))) ≤ |
| (Module.finrank ℂ V : ℝ)^((u+2)/(b : ℝ)) * ∑ i, inverseDegreeSum (H i) u := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro i _ |
| have ht := smallDegreeSum_le_inverseDegreeSum (H i) hu (Real.rpow_pos_of_pos hD (1/(b : ℝ))) |
| rw [← Real.rpow_mul hD.le] at ht |
| convert ht using 1; congr 2; ring |
| have hn := ht.trans (mul_le_mul_of_nonneg_left hs (mul_nonneg hB hm)) |
| apply (mul_le_mul_iff_right₀ hD).mp |
| calc |
| _ ≤ _ := hn |
| _ = _ := by |
| rw [show (u+2)/(b : ℝ) = (-1+(u+2)/(b : ℝ))+1 by ring, |
| Real.rpow_add hD, Real.rpow_one] |
| ring_nf |
| |
| end |
| end PartialPermutation |
| |
| end |
| |
| end OAI |
| |
