| import Mathlib |
| import OAI.Probability.BinarySweep.FiniteLaws.LinearEvenMoment |
| import OAI.Probability.BinarySweep.YoungTheory.HookInducedMoment |
| import OAI.Probability.BinarySweep.MatrixBounds.BlockTrace |
|
|
| namespace OAI |
|
|
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps.Irrep |
| open TraceHolder |
|
|
| variable {I W : Type*} [Fintype I] [DecidableEq I] |
| [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W] |
| local notation "HW" => CoindHilbertSpace (I:=I) (W:=W) |
|
|
| omit [DecidableEq I] [FiniteDimensional ℂ W] in |
| lemma fiber_sum (v : HW) : ∑i, fiberInclude i (v i)=v := by |
| classical |
| apply (WithLp.linearEquiv 2 ℂ (I → W)).injective |
| simp only [map_sum] |
| ext i |
| simp only [fiberInclude,LinearMap.comp_apply,LinearEquiv.coe_coe, |
| LinearEquiv.apply_symm_apply,LinearMap.single_apply,Finset.sum_apply,Pi.single_apply] |
| simp only [WithLp.coe_linearEquiv] |
| let : DecidableEq I := fun i j => Classical.propDecidable (i=j) |
| exact (Finset.sum_ite_eq Finset.univ i (fun x => v x)).trans (ite_eq_left (Finset.mem_univ i)) |
|
|
| omit [DecidableEq I] [FiniteDimensional ℂ W] in |
| lemma fiber_identity : (∑i : I, (fiberInclude i).comp (fiberProject i))=(1 : HW →ₗ[ℂ] HW) := by |
| apply LinearMap.ext |
| intro v |
| simpa only [LinearMap.sum_apply,LinearMap.comp_apply, fiberProject, Module.End.one_apply, |
| LinearEquiv.coe_coe, LinearMap.proj_apply, WithLp.coe_linearEquiv] using fiber_sum v |
|
|
| omit [DecidableEq I] [FiniteDimensional ℂ W] in |
| lemma hilbertBlock_mul (A B : HW →ₗ[ℂ] HW) (i j : I) : |
| hilbertBlock (A*B) i j = ∑a, hilbertBlock A i a * hilbertBlock B a j := by |
| ext v |
| change A (B (fiberInclude j v)) i = _ |
| conv_lhs => rw [← fiber_sum (B (fiberInclude j v))] |
| simp only [map_sum,LinearMap.sum_apply,Module.End.mul_apply] |
| change (WithLp.linearEquiv 2 ℂ (I → W)) (∑ x, A (fiberInclude x (B (fiberInclude j v) x))) i = _ |
| rw [map_sum,Finset.sum_apply] |
| rfl |
|
|
| omit [Fintype I] [FiniteDimensional ℂ W] in |
| lemma hilbertBlock_one (i j : I) : hilbertBlock (1 : HW →ₗ[ℂ] HW) i j = if i=j then 1 else 0 := by |
| ext v |
| change fiberInclude j v i = _ |
| rw [fiberInclude_apply] |
| split_ifs <;> rfl |
| |
| omit [Fintype I] [DecidableEq I] [FiniteDimensional ℂ W] in |
| lemma hilbertBlock_sum {J : Type*} (s : Finset J) (A : J → HW →ₗ[ℂ] HW) (i j : I) : |
| hilbertBlock (∑a∈s,A a) i j = ∑a∈s,hilbertBlock (A a) i j := by |
| ext v |
| simp [hilbertBlock,LinearMap.sum_apply,LinearMap.comp_apply,map_sum] |
|
|
| omit [Fintype I] [DecidableEq I] [FiniteDimensional ℂ W] in |
| lemma hilbertBlock_smul (r : ℝ) (A : HW →ₗ[ℂ] HW) (i j : I) : |
| hilbertBlock (r • A) i j=r • hilbertBlock A i j := by |
| ext v |
| simp [hilbertBlock] |
|
|
| lemma fiber_adjoint (i : I) : (fiberInclude i : W →ₗ[ℂ] HW).adjoint=fiberProject i := by |
| symm |
| apply (LinearMap.eq_adjoint_iff _ _).mpr |
| intro x y |
| simp only [PiLp.inner_apply,fiberInclude_apply,apply_ite,inner_zero_right] |
| change _ = ∑j, if j=i then inner ℂ (x j) y else 0 |
| simp only [fiberProject,LinearMap.comp_apply,LinearMap.proj_apply,LinearEquiv.coe_coe, |
| WithLp.coe_linearEquiv,Finset.sum_ite_eq',Finset.mem_univ,ite_true] |
|
|
| lemma hilbertBlock_adjoint (A : HW →ₗ[ℂ] HW) (i j : I) : |
| hilbertBlock A.adjoint i j=(hilbertBlock A j i).adjoint := by |
| unfold hilbertBlock |
| rw [LinearMap.adjoint_comp,LinearMap.adjoint_comp,fiber_adjoint, |
| ← fiber_adjoint (W:=W) j,LinearMap.adjoint_adjoint] |
| rfl |
|
|
| def hilbertBlocksHom : (HW →ₗ[ℂ] HW) →* Matrix I I (W →ₗ[ℂ] W) where |
| toFun A := hilbertBlock A |
| map_one' := by funext i j; exact hilbertBlock_one i j |
| map_mul' A B := by funext i j; exact hilbertBlock_mul A B i j |
| |
| omit [DecidableEq I] in |
| lemma trace_hilbertBlocks (A : HW →ₗ[ℂ] HW) : |
| LinearMap.trace ℂ HW A = ∑i, LinearMap.trace ℂ W (hilbertBlock A i i) := by |
| conv_lhs => rw [← mul_one A,← fiber_identity] |
| rw [Finset.mul_sum,map_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| change LinearMap.trace ℂ HW ((A.comp (fiberInclude i)).comp (fiberProject i)) = _ |
| rw [LinearMap.trace_comp_comm'] |
| rfl |
|
|
| lemma trace_hilbertBlocks_prod {n : ℕ} (A : Fin (n+1) → HW →ₗ[ℂ] HW) : |
| LinearMap.trace ℂ HW (List.ofFn A).prod = |
| ∑x : Fin (n+1) → I, LinearMap.trace ℂ W |
| (List.ofFn (fun j => hilbertBlock (A j) (x j) (x (finRotate (n+1) j)))).prod := by |
| rw [trace_hilbertBlocks] |
| rw [← map_sum] |
| change LinearMap.trace ℂ W (Matrix.trace (hilbertBlocksHom (I:=I) (W:=W) (List.ofFn A).prod)) = _ |
| have hm := (hilbertBlocksHom (I:=I) (W:=W)).map_list_prod (List.ofFn A) |
| rw [List.map_ofFn] at hm |
| rw [hm,trace_prod_eq_sum_cycles,map_sum] |
| rfl |
|
|
| theorem hilbert_block_trace_holder {q : ℕ} (hq : 0<q) (A : Fin (2*q) → HW →ₗ[ℂ] HW) : |
| ‖LinearMap.trace ℂ HW (List.ofFn A).prod‖ ≤ |
| ∑x : Fin (2*q) → I, ∏j, |
| evenMoment q (hilbertBlock (A j) (x j) (x (finRotate (2*q) j)))^(((2*q:ℕ):ℝ)⁻¹) := by |
| have ht : LinearMap.trace ℂ HW (List.ofFn A).prod = |
| ∑x : Fin (2*q) → I, LinearMap.trace ℂ W |
| (List.ofFn (fun j => hilbertBlock (A j) (x j) (x (finRotate (2*q) j)))).prod := by |
| generalize hm : 2*q=m at A ⊢ |
| cases m with |
| | zero => omega |
| | succ n => exact trace_hilbertBlocks_prod A |
| rw [ht] |
| exact (norm_sum_le _ _).trans (Finset.sum_le_sum (fun _ _ => linear_trace_holder hq _)) |
|
|
| theorem evenMoment_blocks_le {q : ℕ} (hq : 0<q) (K : HW →ₗ[ℂ] HW) : |
| evenMoment q K ≤ ∑x : Fin (2*q) → I, ∏j, |
| (if j.val%2=0 then |
| evenMoment q (hilbertBlock K (x (finRotate (2*q) j)) (x j)) |
| else evenMoment q (hilbertBlock K (x j) (x (finRotate (2*q) j))))^(((2*q:ℕ):ℝ)⁻¹) := by |
| have ht := hilbert_block_trace_holder hq |
| (fun j : Fin (2*q) => if j.val%2=0 then K.adjoint else K) |
| rw [alternating_prod] at ht |
| refine (Complex.re_le_norm _).trans (ht.trans_eq ?_) |
| apply Finset.sum_congr rfl |
| intro x _ |
| apply Finset.prod_congr rfl |
| intro j _ |
| split_ifs <;> simp only [hilbertBlock_adjoint,evenMoment_adjoint] |
| |
| end BinaryCoordinateSweeps.Irrep |
| |
| end |
| |
| end OAI |
| |
