| import Mathlib |
| import OAI.RepresentationTheory.PartialPermutation.DeficitPlancherel |
| import OAI.RepresentationTheory.PartialPermutation.NormBounds |
|
|
| namespace OAI |
|
|
| namespace PartialPermutation |
| noncomputable section |
| open scoped Classical |
|
|
| theorem complementaryCosets_finite : ComplementaryCosetsFiniteStatement := by |
| intro X _ _ b hb M hnonempty hdis _ V _ _ _ ρ hirr hρ f B u hf hm hu hcap |
| let := hirr |
| have hB : 0 ≤ B := by |
| let i : Fin b := ⟨0,hb⟩ |
| have hp : (0:ℝ) < (blockSubgroup (M i)).index := by |
| exact_mod_cast Nat.pos_of_ne_zero (blockSubgroup (M i)).index_ne_zero_of_finite |
| have hz : 0 ≤ B / (blockSubgroup (M i)).index := |
| (Finset.sum_nonneg (fun (g : blockSubgroup (M i)) _ => hf (1*g))).trans (hcap i 1) |
| simpa only [div_mul_cancel₀ B hp.ne'] using mul_nonneg hz hp.le |
| have hsum : (∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u) ≤ |
| (b:ℝ)*symmetricDegreeConstant u := by |
| calc |
| _ ≤ ∑ _i : Fin b, symmetricDegreeConstant u := by |
| apply Finset.sum_le_sum |
| intro i _ |
| let : Nonempty (M i) := (hnonempty i).to_subtype |
| exact symmetricInverseDegreeSum_le_constant (by exact Fintype.card_pos) hu |
| _ = _ := by simp |
| have hs : 0 ≤ ∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u := by |
| exact Finset.sum_nonneg (fun _ _ => inverseDegreeSum_nonneg _ _) |
| have hR : 0 ≤ (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b:ℝ)) := Real.rpow_nonneg (by positivity) _ |
| have hcoef : B * mass f * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) ≤ |
| B * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) := by |
| calc |
| _ ≤ B * 1 * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) := |
| mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hm hB) hR |
| _ = _ := by ring |
| have hHS : hsNormSq (complexFourier ρ (fun g => f g)) ≤ |
| b * B * symmetricDegreeConstant u * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) := by |
| calc |
| _ ≤ _ := by |
| have h := inverse_degree_sum_transfer ρ (fun i => blockSubgroup (M i)) hb hρ |
| (fun i j hij g hg k hk => blockSubgroup_commute (hdis hij) hg hk) f B u hf hu hcap |
| rw [inverseDegreeSum_block_sum M u] at h |
| exact h |
| _ ≤ (B * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ))) * |
| ((b:ℝ)*symmetricDegreeConstant u) := |
| mul_le_mul hcoef hsum hs (mul_nonneg hB hR) |
| _ = _ := by ring |
| exact ⟨hHS,(opNorm_sq_le_hsNormSq _).trans hHS⟩ |
|
|
| end |
| end PartialPermutation |
|
|
| end OAI |
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