import Mathlib import OAI.RepresentationTheory.SignedTensor.Domination namespace OAI noncomputable section open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder attribute [local instance] Classical.propDecidable noncomputable section open scoped BigOperators ComplexConjugate Matrix.Norms.L2Operator attribute [local instance] Classical.propDecidable noncomputable section open scoped BigOperators ComplexOrder MatrixOrder attribute [local instance] Classical.propDecidable noncomputable section open scoped BigOperators ENNReal open MeasureTheory namespace CoordinateSweeps.SignedTensor variable {B : Type} [Fintype B] [DecidableEq B] /- Tensorization of finite density domination, preserving finite probability normalization and the exact product of local dimensions. -/ theorem finite_block_density_domination (s p : ℕ) (hp : 0 < p) (hps : p ≤ s) (l : B → ℕ) (hl : ∀ i, l i ≤ s) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (l i)))) : ∃ (K : Type) (_ : Fintype K) (_ : Nonempty K) (r : K → B → EvenDensity p), (((∏ i, ((ρ i).dimension : ℂ))*((s+1 : ℕ) : ℂ)^(Fintype.card B*(41*p^2))) • ((Fintype.card K : ℂ)⁻¹ • ∑ x, DependentTensor.tensor (fun i => densityPower (l i) (r x i).val))- DependentTensor.tensor (fun i => isotypic (p := p) (ρ i))).PosSemidef := by choose K fK nK r hr using fun i => finite_density_domination s p (l i) hp hps (hl i) (ρ i) let : ∀ i, Fintype (K i) := fK let : ∀ i, Nonempty (K i) := nK refine ⟨(∀ i, K i),inferInstance,inferInstance,fun x i => r i (x i),?_⟩ have hh := DependentTensor.tensor_mono (fun i => isotypic (p := p) (ρ i)) (fun i => (((ρ i).dimension : ℂ)*((s+1 : ℕ) : ℂ)^(41*p^2)) • ((Fintype.card (K i) : ℂ)⁻¹ • ∑ x, densityPower (l i) (r i x).val)) (fun i => isotypic_posSemidef (ρ i)) hr rw [DependentTensor.tensor_smul] at hh have hm : DependentTensor.tensor (fun i => (Fintype.card (K i) : ℂ)⁻¹ • ∑ x, densityPower (l i) (r i x).val) = (Fintype.card (∀ i, K i) : ℂ)⁻¹ • ∑ x : ∀ i, K i, DependentTensor.tensor (fun i => densityPower (l i) (r i (x i)).val) := by simp only [Finset.smul_sum] rw [DependentTensor.tensor_mixture] rw [Fintype.card_pi,Nat.cast_prod,Finset.prod_inv_distrib] have hc : (∏ i, ((ρ i).dimension : ℂ)*((s+1 : ℕ) : ℂ)^(41*p^2)) = (∏ i, ((ρ i).dimension : ℂ))*((s+1 : ℕ) : ℂ)^(Fintype.card B*(41*p^2)) := by rw [Finset.prod_mul_distrib,Finset.prod_const,Finset.card_univ,← pow_mul,mul_comm (41*p^2)] rwa [hm,hc] at hh end CoordinateSweeps.SignedTensor noncomputable section open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder noncomputable section open scoped BigOperators attribute [local instance] Classical.propDecidable namespace CoordinateSweeps.OrderedBlocks variable {r : ℕ} (n : Fin r → ℕ) def offset (i : Fin r) : ℕ := ∑ j ∈ Finset.Iio i, n j lemma offset_next_le (i j : Fin r) (hij : i < j) : offset n i+n i ≤ offset n j := by have he : Finset.Iic i=insert i (Finset.Iio i) := by ext x; simp [le_iff_lt_or_eq,or_comm] have hh : ∑ k ∈ Finset.Iic i, n k ≤ ∑ k ∈ Finset.Iio j, n k := Finset.sum_le_sum_of_subset (fun k hk => Finset.mem_Iio.mpr (lt_of_le_of_lt (Finset.mem_Iic.mp hk) hij)) rw [he,Finset.sum_insert (by simp)] at hh simpa only [offset,add_comm] using hh lemma offset_next_le_total (i : Fin r) : offset n i+n i ≤ ∑ j, n j := by have he : Finset.Iic i=insert i (Finset.Iio i) := by ext x; simp [le_iff_lt_or_eq,or_comm] have hh : ∑ k ∈ Finset.Iic i, n k ≤ ∑ k, n k := Finset.sum_le_sum_of_subset (Finset.subset_univ _) rw [he,Finset.sum_insert (by simp)] at hh simpa only [offset,add_comm] using hh def encode (x : (i : Fin r) × Fin (n i)) : Fin (∑ i, n i) := ⟨offset n x.1+x.2,lt_of_lt_of_le (Nat.add_lt_add_left x.2.isLt _) (offset_next_le_total n x.1)⟩ lemma encode_lt_of_block {i j : Fin r} (hij : i < j) (a : Fin (n i)) (b : Fin (n j)) : encode n ⟨i,a⟩ < encode n ⟨j,b⟩ := by change offset n i+a.val < offset n j+b.val have hh := offset_next_le n i j hij have ha := a.isLt omega lemma encode_lt_same {i : Fin r} (a b : Fin (n i)) : encode n ⟨i,a⟩ < encode n ⟨i,b⟩ ↔ a < b := by change offset n i+a.val < offset n i+b.val ↔ a.val < b.val omega lemma encode_injective : Function.Injective (encode n) := by rintro ⟨i,a⟩ ⟨j,b⟩ h have hi : i=j := by rcases lt_trichotomy i j with hij | hij | hij · exact False.elim ((ne_of_lt (encode_lt_of_block n hij a b)) h) · exact hij · exact False.elim ((ne_of_lt (encode_lt_of_block n hij b a)) h.symm) subst j have hab : a=b := Fin.ext (by have hh := congrArg Fin.val h; dsimp [encode] at hh; omega) subst b rfl /- Explicit contiguous coordinate chart, including zero-length blocks. -/ def positions : ((i : Fin r) × Fin (n i)) ≃ Fin (∑ i, n i) := Equiv.ofBijective (encode n) ((Fintype.bijective_iff_injective_and_card _).mpr ⟨encode_injective n,by simp [Fintype.card_sigma]⟩) lemma positions_apply (x : (i : Fin r) × Fin (n i)) : positions n x=encode n x := rfl lemma positions_lt_of_block {i j : Fin r} (hij : i < j) (a : Fin (n i)) (b : Fin (n j)) : positions n ⟨i,a⟩ < positions n ⟨j,b⟩ := encode_lt_of_block n hij a b lemma positions_lt_same {i : Fin r} (a b : Fin (n i)) : positions n ⟨i,a⟩ < positions n ⟨i,b⟩ ↔ a < b := encode_lt_same n a b def perm (σ : ∀ i, Equiv.Perm (Fin (n i))) : Equiv.Perm (Fin (∑ i, n i)) := (positions n).permCongr (Equiv.sigmaCongrRight σ) @[simp] lemma perm_apply (σ : ∀ i, Equiv.Perm (Fin (n i))) (i : Fin r) (j : Fin (n i)) : perm n σ (positions n ⟨i,j⟩)=positions n ⟨i,σ i j⟩ := by simp [perm,Equiv.permCongr_apply] lemma perm_inv (σ : ∀ i, Equiv.Perm (Fin (n i))) : (perm n σ)⁻¹=perm n (fun i => (σ i)⁻¹) := by apply Equiv.ext intro x obtain ⟨⟨i,j⟩,rfl⟩ := (positions n).surjective x apply (perm n σ).injective simp def permHom : (∀ i, Equiv.Perm (Fin (n i))) →* Equiv.Perm (Fin (∑ i, n i)) where toFun := perm n map_one' := by apply Equiv.ext; intro x obtain ⟨⟨i,j⟩,rfl⟩ := (positions n).surjective x simp map_mul' σ τ := by apply Equiv.ext; intro x obtain ⟨⟨i,j⟩,rfl⟩ := (positions n).surjective x simp end CoordinateSweeps.OrderedBlocks namespace CoordinateSweeps.SignedTensor lemma phase_double_prod {p l : ℕ} (σ : Equiv.Perm (Fin l)) (w : Word p l) : phase σ w=∏ i : Fin l, ∏ j : Fin l, if j < i ∧ odd (w i) ∧ odd (w j) ∧ σ i < σ j then (-1 : ℂ) else 1 := by have he : Equiv.Perm.finPairsLT l=(Finset.univ : Finset ((i : Fin l) × Fin l)).filter (fun x => x.2 < x.1) := by ext x simp only [Equiv.Perm.mem_finPairsLT,Finset.mem_filter,Finset.mem_univ,true_and] rw [phase_prod,he,Finset.prod_filter,Fintype.prod_sigma] apply Finset.prod_congr rfl intro i hi apply Finset.prod_congr rfl intro j hj split_ifs <;> simp_all def blockPerm {m n : ℕ} (σ : Equiv.Perm (Fin m)) (τ : Equiv.Perm (Fin n)) : Equiv.Perm (Fin (m+n)) := finSumFinEquiv.permCongr (Equiv.Perm.sumCongr σ τ) @[simp] lemma blockPerm_left {m n : ℕ} (σ : Equiv.Perm (Fin m)) (τ : Equiv.Perm (Fin n)) (i : Fin m) : blockPerm σ τ (Fin.castAdd n i)=Fin.castAdd n (σ i) := by simp [blockPerm,Equiv.permCongr_apply] @[simp] lemma blockPerm_right {m n : ℕ} (σ : Equiv.Perm (Fin m)) (τ : Equiv.Perm (Fin n)) (i : Fin n) : blockPerm σ τ (Fin.natAdd m i)=Fin.natAdd m (τ i) := by simp [blockPerm,Equiv.permCongr_apply] lemma blockPerm_inv {m n : ℕ} (σ : Equiv.Perm (Fin m)) (τ : Equiv.Perm (Fin n)) : (blockPerm σ τ)⁻¹=blockPerm σ⁻¹ τ⁻¹ := by apply Equiv.ext intro i apply (blockPerm σ τ).injective induction i using Fin.addCases <;> simp lemma phase_blockPerm {p m n : ℕ} (σ : Equiv.Perm (Fin m)) (τ : Equiv.Perm (Fin n)) (w : Word p (m+n)) : phase (blockPerm σ τ) w=phase σ (fun i => w (Fin.castAdd n i))* phase τ (fun i => w (Fin.natAdd m i)) := by have hlr (i : Fin m) (j : Fin n) : ¬ Fin.natAdd m j w (Fin.castAdd n i),fun i => w (Fin.natAdd m i)) invFun w := Fin.addCases w.1 w.2 left_inv w := by funext i; induction i using Fin.addCases <;> simp right_inv w := by ext <;> simp lemma splitWord_wordPerm {p m n : ℕ} (σ : Equiv.Perm (Fin m)) (τ : Equiv.Perm (Fin n)) (w : Word p (m+n)) : splitWord p m n (wordPerm (blockPerm σ τ) w)= (wordPerm σ (splitWord p m n w).1,wordPerm τ (splitWord p m n w).2) := by ext i <;> simp [splitWord,wordPerm,blockPerm_inv] def blockMatrix {p m n : ℕ} (A : Matrix (Word p m) (Word p m) ℂ) (B : Matrix (Word p n) (Word p n) ℂ) : Matrix (Word p (m+n)) (Word p (m+n)) ℂ := fun u w => A (splitWord p m n u).1 (splitWord p m n w).1 * B (splitWord p m n u).2 (splitWord p m n w).2 /- Exact block multiplicativity of the signed permutation action. -/ theorem matrix_blockPerm {p m n : ℕ} (σ : Equiv.Perm (Fin m)) (τ : Equiv.Perm (Fin n)) : matrix (p := p) (blockPerm σ τ)=blockMatrix (matrix σ) (matrix τ) := by ext u w have he : u=wordPerm (blockPerm σ τ) w ↔ (splitWord p m n u).1=wordPerm σ (splitWord p m n w).1 ∧ (splitWord p m n u).2=wordPerm τ (splitWord p m n w).2 := by rw [← (splitWord p m n).injective.eq_iff,splitWord_wordPerm,Prod.mk.injEq] simp only [matrix,blockMatrix,he,phase_blockPerm] split_ifs <;> simp_all [splitWord] end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor open OrderedBlocks variable {p r : ℕ} (n : Fin r → ℕ) lemma prod_positions (f : Fin (∑ i, n i) → ℂ) : ∏ x, f x=∏ i, ∏ j : Fin (n i), f (positions n ⟨i,j⟩) := by rw [← Equiv.prod_comp (positions n) f,Fintype.prod_sigma] lemma phase_grouped (σ : ∀ i, Equiv.Perm (Fin (n i))) (w : Word p (∑ i, n i)) : phase (perm n σ) w=∏ i, phase (σ i) (fun j => w (positions n ⟨i,j⟩)) := by rw [phase_double_prod,prod_positions n] apply Finset.prod_congr rfl intro i hi rw [phase_double_prod] apply Finset.prod_congr rfl intro a ha rw [prod_positions n,Finset.prod_eq_single i] · apply Finset.prod_congr rfl intro b hb simp only [perm_apply,positions_lt_same] · intro j hj hji apply Finset.prod_eq_one intro b hb rcases lt_or_gt_of_ne hji with hlt | hgt · have hn : ¬ positions n ⟨i,σ i a⟩ < positions n ⟨j,σ j b⟩ := not_lt_of_ge (positions_lt_of_block n hlt _ _).le simp [hn] · have hn : ¬ positions n ⟨j,b⟩ < positions n ⟨i,a⟩ := not_lt_of_ge (positions_lt_of_block n hgt _ _).le simp [hn] · simp def groupWord (p : ℕ) : Word p (∑ i, n i) ≃ (∀ i, Word p (n i)) where toFun w i j := w (positions n ⟨i,j⟩) invFun w x := w ((positions n).symm x).1 ((positions n).symm x).2 left_inv w := by funext x; simp right_inv w := by funext i j change (fun x : (i : Fin r) × Fin (n i) => w x.1 x.2) ((positions n).symm (positions n ⟨i,j⟩)) = _ rw [Equiv.symm_apply_apply] lemma groupWord_wordPerm (σ : ∀ i, Equiv.Perm (Fin (n i))) (w : Word p (∑ i, n i)) : groupWord n p (wordPerm (perm n σ) w)=fun i => wordPerm (σ i) (groupWord n p w i) := by funext i j simp [groupWord,wordPerm,perm_inv] /- Genuine product-subgroup action, with no suppressed Koszul signs. -/ theorem matrix_grouped (σ : ∀ i, Equiv.Perm (Fin (n i))) : matrix (p := p) (perm n σ)= Matrix.reindex (groupWord n p).symm (groupWord n p).symm (DependentTensor.tensor (fun i => matrix (σ i))) := by ext u w have he : u=wordPerm (perm n σ) w ↔ ∀ i, groupWord n p u i=wordPerm (σ i) (groupWord n p w i) := by rw [← (groupWord n p).injective.eq_iff,groupWord_wordPerm] exact funext_iff change (if u=wordPerm (perm n σ) w then phase (perm n σ) w else 0)= ∏ i, if groupWord n p u i=wordPerm (σ i) (groupWord n p w i) then phase (σ i) (groupWord n p w i) else 0 simp only [he] by_cases h : ∀ i, groupWord n p u i=wordPerm (σ i) (groupWord n p w i) · rw [ite_eq_left h,phase_grouped] simp only [h,ite_true] rfl · rw [ite_eq_right h] obtain ⟨i,hi⟩ := not_forall.mp h symm apply Finset.prod_eq_zero (Finset.mem_univ i) simp [hi] end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor open OrderedBlocks UnitaryIrrep variable {p r : ℕ} (n : Fin r → ℕ) /- The subgroup operator is an actual group-algebra average on the full set of slots, not an operator posited to factor. -/ def groupedAverage {Ω : Fin r → Type*} [∀ i, Fintype (Ω i)] (w : ∀ i, Ω i → ℂ) (g : ∀ i, Ω i → Equiv.Perm (Fin (n i))) : Matrix (Word p (∑ i, n i)) (Word p (∑ i, n i)) ℂ := averageMatrix (action p (∑ i, n i)) (fun ω : ∀ i, Ω i => ∏ i, w i (ω i)) (fun ω => perm n (fun i => g i (ω i))) lemma groupedAverage_factor {Ω : Fin r → Type*} [∀ i, Fintype (Ω i)] (w : ∀ i, Ω i → ℂ) (g : ∀ i, Ω i → Equiv.Perm (Fin (n i))) : groupedAverage (p := p) n w g= Matrix.reindex (groupWord n p).symm (groupWord n p).symm (DependentTensor.tensor (fun i => averageMatrix (action p (n i)) (w i) (g i))) := by ext u v simp only [groupedAverage,averageMatrix,Matrix.reindex_apply,Matrix.submatrix_apply, Equiv.symm_symm,DependentTensor.tensor,Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul] change (∑ ω : ∀ i, Ω i, (∏ i, w i (ω i))* matrix (perm n (fun i => g i (ω i))) u v)= ∏ i, ∑ ω : Ω i, w i ω*matrix (g i ω) (groupWord n p u i) (groupWord n p v i) have he (σ : ∀ i, Equiv.Perm (Fin (n i))) : matrix (perm n σ) u v=∏ i, matrix (σ i) (groupWord n p u i) (groupWord n p v i) := by have hh := congrArg (fun A : Matrix (Word p (∑ i, n i)) (Word p (∑ i, n i)) ℂ => A u v) (matrix_grouped (p := p) n σ) exact hh simp only [he] rw [Fintype.prod_sum] apply Finset.sum_congr rfl intro ω hω exact (Finset.prod_mul_distrib : (∏ i, w i (ω i)*matrix (g i (ω i)) (groupWord n p u i) (groupWord n p v i))=_).symm def groupedProjector (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) : Matrix (Word p (∑ i, n i)) (Word p (∑ i, n i)) ℂ := groupedAverage n (fun i => (ρ i).coefficient) (fun _ => id) lemma groupedProjector_factor (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) : groupedProjector (p := p) n ρ= Matrix.reindex (groupWord n p).symm (groupWord n p).symm (DependentTensor.tensor (fun i => (ρ i).projector (action p (n i)))) := by exact groupedAverage_factor n _ _ lemma matrix_trace_reindex {X Y : Type*} [Fintype X] [Fintype Y] (e : X ≃ Y) (A : Matrix X X ℂ) : Matrix.trace (Matrix.reindex e e A)=Matrix.trace A := by exact Equiv.sum_comp e.symm (fun i => A i i) lemma reindex_moment {X Y : Type*} [Fintype X] [Fintype Y] [DecidableEq X] [DecidableEq Y] (e : X ≃ Y) (A : Matrix X X ℂ) (q : ℕ) : Matrix.trace ((((Matrix.reindexAlgEquiv ℂ ℂ e A).conjTranspose)*(Matrix.reindexAlgEquiv ℂ ℂ e A))^q)= Matrix.trace ((A.conjTranspose*A)^q) := by have hs : (Matrix.reindexAlgEquiv ℂ ℂ e A).conjTranspose= Matrix.reindexAlgEquiv ℂ ℂ e A.conjTranspose := by ext i j; rfl rw [hs,← map_mul,← map_pow] exact matrix_trace_reindex e _ /- Exact isotypic child moment factorization, including the true Hom multiplicities and all zero-length blocks. -/ theorem grouped_projected_moment {Ω : Fin r → Type*} [∀ i, Fintype (Ω i)] (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) (w : ∀ i, Ω i → ℂ) (g : ∀ i, Ω i → Equiv.Perm (Fin (n i))) {q : ℕ} (hq : q ≠ 0) : Matrix.trace (((groupedProjector (p := p) n ρ*groupedAverage n w g).conjTranspose* (groupedProjector n ρ*groupedAverage n w g))^q)= ∏ i, (Module.finrank ℂ ((ρ i).asRepresentation.IntertwiningMap (matrixRepresentation (action p (n i)))) : ℂ)* Matrix.trace (((averageMatrix (ρ i).matrix (w i) (g i)).conjTranspose* averageMatrix (ρ i).matrix (w i) (g i))^q) := by rw [groupedProjector_factor,groupedAverage_factor] let e := (groupWord n p).symm change Matrix.trace (((Matrix.reindexAlgEquiv ℂ ℂ e _*Matrix.reindexAlgEquiv ℂ ℂ e _).conjTranspose* (Matrix.reindexAlgEquiv ℂ ℂ e _*Matrix.reindexAlgEquiv ℂ ℂ e _))^q)=_ rw [← map_mul,reindex_moment,← DependentTensor.tensor_mul,DependentTensor.tensor_moment] apply Finset.prod_congr rfl intro i hi exact (ρ i).projected_operator_moment _ (fun σ => (matrix_inv σ).symm) (w i) (g i) hq end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor variable {p : ℕ} def IsEven (A : Matrix (Letter p) (Letter p) ℂ) : Prop := ∀ i j, (odd i ↔ ¬ odd j) → A i j=0 def grading (p : ℕ) : Matrix (Letter p) (Letter p) ℂ := Matrix.diagonal (fun i => if odd i then 1 else 0) lemma isEven_iff_commute (A : Matrix (Letter p) (Letter p) ℂ) : IsEven A ↔ Commute A (grading p) := by constructor · intro h show A*grading p=grading p*A ext i j simp only [grading,Matrix.mul_diagonal,Matrix.diagonal_mul] by_cases hi : odd i · by_cases hj : odd j · simp [hi,hj] · simp [hi,hj,h i j (by tauto)] · by_cases hj : odd j · simp [hi,hj,h i j (by tauto)] · simp [hi,hj] · intro h i j hij have hh := congrArg (fun M : Matrix (Letter p) (Letter p) ℂ => M i j) h.eq simp only [grading,Matrix.mul_diagonal,Matrix.diagonal_mul] at hh by_cases hi : odd i · have hj : ¬ odd j := hij.mp hi simpa [hi,hj] using hh.symm · have hj : odd j := by tauto simpa [hi,hj] using hh lemma isEven_add {A B : Matrix (Letter p) (Letter p) ℂ} (hA : IsEven A) (hB : IsEven B) : IsEven (A+B) := by intro i j hij; simp only [Matrix.add_apply,hA i j hij,hB i j hij,add_zero] lemma isEven_smul (c : ℂ) {A : Matrix (Letter p) (Letter p) ℂ} (hA : IsEven A) : IsEven (c • A) := by intro i j hij; simp only [Matrix.smul_apply,smul_eq_mul,hA i j hij,mul_zero] lemma isEven_sum {I : Type*} [Fintype I] (A : I → Matrix (Letter p) (Letter p) ℂ) (hA : ∀ i, IsEven (A i)) : IsEven (∑ i, A i) := by intro i j hij simp only [Matrix.sum_apply] exact Finset.sum_eq_zero (fun a _ => hA a i j hij) lemma isEven_one : IsEven (1 : Matrix (Letter p) (Letter p) ℂ) := by intro i j hij apply Matrix.one_apply_ne intro he subst j tauto lemma densityPower_commute_of_even {l : ℕ} (r : Matrix (Letter p) (Letter p) ℂ) (hr : IsEven r) (σ : Equiv.Perm (Fin l)) : matrix σ*densityPower l r=densityPower l r*matrix σ := by have hpar {u w : Word p l} (h : densityPower l r u w ≠ 0) : ∀ i, odd (u i) ↔ odd (w i) := by intro i have he : r (u i) (w i) ≠ 0 := (Finset.prod_ne_zero_iff.mp h) i (Finset.mem_univ _) by_contra hn exact he (hr _ _ (by tauto)) ext u w simp only [Matrix.mul_apply,matrix] rw [Finset.sum_eq_single ((wordPerm σ).symm u),Finset.sum_eq_single (wordPerm σ w)] · simp only [(wordPerm σ).apply_symm_apply,ite_true] have hd : densityPower l r u (wordPerm σ w)=densityPower l r ((wordPerm σ).symm u) w := by simpa using densityPower_permute σ r ((wordPerm σ).symm u) w rw [hd] by_cases hn : densityPower l r ((wordPerm σ).symm u) w=0 · simp [hn] · rw [phase_eq_of_parity σ _ _ (hpar hn),mul_comm] · intro v hv hne; simp [hne] · simp · intro v hv hne have hh : u ≠ wordPerm σ v := by intro he exact hne ((Equiv.eq_symm_apply _).mpr he.symm) simp [hh] · simp lemma densityPower_commute_lift {l : ℕ} (r : Matrix (Letter p) (Letter p) ℂ) (hr : IsEven r) (x : MonoidAlgebra ℂ (Equiv.Perm (Fin l))) : (MonoidAlgebra.lift ℂ _ _ (action p l) x)*densityPower l r = densityPower l r*(MonoidAlgebra.lift ℂ _ _ (action p l) x) := by induction x using MonoidAlgebra.induction_on with | of g => simpa [action] using densityPower_commute_of_even r hr g | add x y hx hy => simp only [map_add,add_mul,mul_add,hx,hy] | smul c x hx => simp only [map_smul,smul_mul_assoc,mul_smul_comm,hx] /- Constructed even whitening of the regularized empirical row density. -/ theorem exists_even_whitening {R : Type*} [Fintype R] [Nonempty R] (r : R → EvenDensity p) {ε : ℝ} (hε : 0 < ε) : let B := (Fintype.card R : ℂ)⁻¹ • ∑ i, (r i).val + (ε : ℂ) • (1 : Matrix (Letter p) (Letter p) ℂ) ∃ A : Matrix (Letter p) (Letter p) ℂ, A.PosSemidef ∧ IsEven A ∧ A*B*A=1 ∧ ((Fintype.card R : ℂ) • (1 : Matrix (Letter p) (Letter p) ℂ)- ∑ i, A*(r i).val*A).PosSemidef := by dsimp only let B := (Fintype.card R : ℂ)⁻¹ • ∑ i, (r i).val + (ε : ℂ) • (1 : Matrix (Letter p) (Letter p) ℂ) have hBp : B.PosDef := TensorBoard.average_regularization_posDef (fun i => (r i).val) (fun i => (r i).property.1) hε have hBe : IsEven B := isEven_add (isEven_smul _ (isEven_sum _ (fun i => (r i).property.2.2))) (isEven_smul _ isEven_one) obtain ⟨A,hA,hABA,hcomm⟩ := TensorBoard.exists_whitening B hBp refine ⟨A,hA,(isEven_iff_commute A).mpr (hcomm _ ((isEven_iff_commute B).mp hBe)),hABA,?_⟩ apply TensorBoard.whitening_sum_bound _ B A hA.isHermitian hABA dsimp [B] convert (Matrix.PosSemidef.one (n := Letter p) (R := ℂ)).smul hε.le using 1 module end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor variable {p l : ℕ} /- Even inhomogeneous site tensors transform by the *signed* permutation matrix with no residual phase. This is what permits regrouping deleted rows and columns without identifying the two Koszul charts by an unsigned map. -/ lemma tensor_covariance (A : Fin l → Matrix (Letter p) (Letter p) ℂ) (hA : ∀ i, IsEven (A i)) (σ : Equiv.Perm (Fin l)) : matrix σ*TensorBoard.tensor A= TensorBoard.tensor (fun i => A (σ⁻¹ i))*matrix σ := by have hpar {u w : Word p l} (h : TensorBoard.tensor A u w ≠ 0) : ∀ i, odd (u i) ↔ odd (w i) := by intro i have he : A i (u i) (w i) ≠ 0 := (Finset.prod_ne_zero_iff.mp h) i (Finset.mem_univ _) by_contra hn exact he (hA i _ _ (by tauto)) ext u w simp only [Matrix.mul_apply,matrix] rw [Finset.sum_eq_single ((wordPerm σ).symm u),Finset.sum_eq_single (wordPerm σ w)] · simp only [(wordPerm σ).apply_symm_apply,ite_true] have hd : TensorBoard.tensor (fun i => A (σ⁻¹ i)) u (wordPerm σ w)= TensorBoard.tensor A ((wordPerm σ).symm u) w := by change (∏ i, A (σ⁻¹ i) (u i) (w (σ⁻¹ i)))=∏ i, A i (u (σ i)) (w i) simpa using (Equiv.prod_comp σ (fun i => A (σ⁻¹ i) (u i) (w (σ⁻¹ i)))).symm rw [hd] by_cases hn : TensorBoard.tensor A ((wordPerm σ).symm u) w=0 · simp [hn] · rw [phase_eq_of_parity σ _ _ (hpar hn),mul_comm] · intro v hv hne; simp [hne] · simp · intro v hv hne have hh : u ≠ wordPerm σ v := by intro he exact hne ((Equiv.eq_symm_apply _).mpr he.symm) simp [hh] · simp lemma tensor_signed_conjugate (A : Fin l → Matrix (Letter p) (Letter p) ℂ) (hA : ∀ i, IsEven (A i)) (σ : Equiv.Perm (Fin l)) : matrix σ*TensorBoard.tensor A*(matrix σ).conjTranspose= TensorBoard.tensor (fun i => A (σ⁻¹ i)) := by rw [tensor_covariance A hA σ,mul_assoc,← matrix_inv,← matrix_mul,mul_inv_cancel,matrix_one,mul_one] end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor open OrderedBlocks variable {p r : ℕ} (n : Fin r → ℕ) def groupedSite (a : Fin r → EvenDensity p) (k : Fin (∑ i, n i)) : EvenDensity p := a ((positions n).symm k).1 lemma grouped_densityPower (a : Fin r → EvenDensity p) : (DependentTensor.tensor (fun i => densityPower (n i) (a i).val)).submatrix (groupWord n p) (groupWord n p)=TensorBoard.tensor (fun k => (groupedSite n a k).val) := by ext u v change (∏ i, ∏ j : Fin (n i), (a i).val (u (positions n ⟨i,j⟩)) (v (positions n ⟨i,j⟩)))= ∏ k, (a ((positions n).symm k).1).val (u k) (v k) rw [prod_positions] apply Finset.prod_congr rfl intro i _ apply Finset.prod_congr rfl intro j _ rw [Equiv.symm_apply_apply] /- Literal contiguous-block projectors are dominated by independent finite mixtures of even site densities, with all local dimension factors. -/ theorem finite_grouped_density_domination (s : ℕ) (hp : 0 < p) (hps : p ≤ s) (hn : ∀ i, n i ≤ s) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) : ∃ (K : Type) (_ : Fintype K) (_ : Nonempty K) (a : K → Fin r → EvenDensity p), (((∏ i, ((ρ i).dimension : ℂ))*((s+1 : ℕ) : ℂ)^(r*(41*p^2))) • ((Fintype.card K : ℂ)⁻¹ • ∑ x, TensorBoard.tensor (fun k => (groupedSite n (a x) k).val))- groupedProjector (p := p) n ρ).PosSemidef := by obtain ⟨K,hK,nK,a,ha⟩ := finite_block_density_domination s p hp hps n hn ρ let := hK let := nK refine ⟨K,hK,nK,a,?_⟩ have hh := ha.submatrix (groupWord n p) have hm : (∑ x, DependentTensor.tensor (fun i => densityPower (n i) (a x i).val)).submatrix (groupWord n p) (groupWord n p)=∑ x, TensorBoard.tensor (fun k => (groupedSite n (a x) k).val) := by ext u v simp only [Matrix.submatrix_apply,Matrix.sum_apply] apply Finset.sum_congr rfl intro x _ exact congrArg (fun M => M u v) (grouped_densityPower n (a x)) simp only [Fintype.card_fin,Matrix.submatrix_sub,Matrix.submatrix_smul, Pi.sub_apply,Pi.smul_apply,hm] at hh have he : (DependentTensor.tensor (fun i => isotypic (p := p) (ρ i))).submatrix (groupWord n p) (groupWord n p)=groupedProjector (p := p) n ρ := by simp only [groupedProjector_factor,isotypic_eq_projector] rfl rwa [he] at hh /- The same domination after an arbitrary *signed* change of slot chart. -/ theorem finite_regrouped_density_domination (s : ℕ) (hp : 0 < p) (hps : p ≤ s) (hn : ∀ i, n i ≤ s) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) (σ : Equiv.Perm (Fin (∑ i, n i))) : ∃ (K : Type) (_ : Fintype K) (_ : Nonempty K) (a : K → Fin r → EvenDensity p), (((∏ i, ((ρ i).dimension : ℂ))*((s+1 : ℕ) : ℂ)^(r*(41*p^2))) • ((Fintype.card K : ℂ)⁻¹ • ∑ x, TensorBoard.tensor (fun k => (groupedSite n (a x) (σ⁻¹ k)).val))- matrix σ*groupedProjector (p := p) n ρ*(matrix σ).conjTranspose).PosSemidef := by obtain ⟨K,hK,nK,a,ha⟩ := finite_grouped_density_domination n s hp hps hn ρ let := hK let := nK refine ⟨K,hK,nK,a,?_⟩ have hh := ha.mul_mul_conjTranspose_same (matrix σ) simp only [mul_sub,sub_mul,Matrix.mul_smul,Matrix.smul_mul,Matrix.mul_sum,Matrix.sum_mul] at hh have ht (x : K) : matrix σ*TensorBoard.tensor (fun k => (groupedSite n (a x) k).val)*(matrix σ).conjTranspose= TensorBoard.tensor (fun k => (groupedSite n (a x) (σ⁻¹ k)).val) := tensor_signed_conjugate _ (fun k => (groupedSite n (a x) k).property.2.2) σ simpa only [ht] using hh end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor variable {p r : ℕ} (n : Fin r → ℕ) lemma groupedProjector_posSemidef (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) : (groupedProjector (p := p) n ρ).PosSemidef := by rw [groupedProjector_factor] exact (DependentTensor.tensor_posSemidef _ (fun i => (ρ i).projector_posSemidef _ (fun σ => (matrix_inv σ).symm))).submatrix _ lemma groupedProjector_idempotent (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) : groupedProjector (p := p) n ρ*groupedProjector n ρ=groupedProjector n ρ := by rw [groupedProjector_factor] change Matrix.reindexAlgEquiv ℂ ℂ (groupWord n p).symm _* Matrix.reindexAlgEquiv ℂ ℂ (groupWord n p).symm _=Matrix.reindexAlgEquiv ℂ ℂ (groupWord n p).symm _ rw [← map_mul,← DependentTensor.tensor_mul] simp only [UnitaryIrrep.projector_idempotent] end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor open OrderedBlocks UnitaryIrrep variable {p r l : ℕ} (n : Fin r → ℕ) def lengthMatrix {a b : ℕ} (h : a=b) (M : Matrix (Word p a) (Word p a) ℂ) : Matrix (Word p b) (Word p b) ℂ := h ▸ M def ambientProjector (h : (∑ i, n i)=l) (σ : Equiv.Perm (Fin l)) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) : Matrix (Word p l) (Word p l) ℂ := matrix σ*lengthMatrix h (groupedProjector n ρ)*(matrix σ).conjTranspose def ambientProjectorAlgebra (h : (∑ i, n i)=l) (σ : Equiv.Perm (Fin l)) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) : MonoidAlgebra ℂ (Equiv.Perm (Fin l)) := averageAlgebra (fun ω : ∀ i, Equiv.Perm (Fin (n i)) => ∏ i, (ρ i).coefficient (ω i)) (fun ω => σ*(Equiv.permCongr (finCongr h) (perm n ω))*σ⁻¹) lemma ambientProjectorAlgebra_lift (h : (∑ i, n i)=l) (σ : Equiv.Perm (Fin l)) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) : MonoidAlgebra.lift ℂ _ _ (action p l) (ambientProjectorAlgebra n h σ ρ)= ambientProjector (p := p) n h σ ρ := by subst l simp only [ambientProjectorAlgebra,lift_averageAlgebra,averageMatrix,ambientProjector, lengthMatrix,groupedProjector,groupedAverage,Matrix.mul_sum,Matrix.sum_mul,Matrix.mul_smul,Matrix.smul_mul] apply Finset.sum_congr rfl intro ω _ congr 1 change matrix (σ*(Equiv.permCongr (finCongr rfl) (perm n ω))*σ⁻¹)=_ simp only [finCongr_refl,matrix_mul,matrix_inv] rfl lemma ambientProjector_posSemidef (h : (∑ i, n i)=l) (σ : Equiv.Perm (Fin l)) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) : (ambientProjector (p := p) n h σ ρ).PosSemidef := by subst l exact (groupedProjector_posSemidef n ρ).mul_mul_conjTranspose_same (matrix σ) lemma ambientProjector_idempotent (h : (∑ i, n i)=l) (σ : Equiv.Perm (Fin l)) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) : ambientProjector (p := p) n h σ ρ*ambientProjector n h σ ρ=ambientProjector n h σ ρ := by subst l unfold ambientProjector lengthMatrix calc _ = matrix σ*(groupedProjector n ρ*((matrix σ).conjTranspose*matrix σ)*groupedProjector n ρ)* (matrix (p := p) σ).conjTranspose := by noncomm_ring _ = _ := by rw [matrix_unitary,mul_one,groupedProjector_idempotent] /- Actual ambient projectors, not hypothesized density mixtures. -/ theorem finite_ambient_density_domination (s : ℕ) (hp : 0 < p) (hps : p ≤ s) (hn : ∀ i, n i ≤ s) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) (h : (∑ i, n i)=l) (σ : Equiv.Perm (Fin l)) : ∃ (K : Type) (_ : Fintype K) (_ : Nonempty K) (a : K → Fin r → EvenDensity p), (((∏ i, ((ρ i).dimension : ℂ))*((s+1 : ℕ) : ℂ)^(r*(41*p^2))) • ((Fintype.card K : ℂ)⁻¹ • ∑ x, TensorBoard.tensor (fun k => (groupedSite n (a x) ((finCongr h).symm (σ⁻¹ k))).val))- ambientProjector (p := p) n h σ ρ).PosSemidef := by subst l simpa [ambientProjector,lengthMatrix] using finite_regrouped_density_domination n s hp hps hn ρ σ end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor open OrderedBlocks UnitaryIrrep variable {p r l : ℕ} (n : Fin r → ℕ) def ambientAverage (h : (∑ i, n i)=l) (σ : Equiv.Perm (Fin l)) {Ω : Fin r → Type*} [∀ i, Fintype (Ω i)] (w : ∀ i, Ω i → ℂ) (g : ∀ i, Ω i → Equiv.Perm (Fin (n i))) : Matrix (Word p l) (Word p l) ℂ := averageMatrix (action p l) (fun ω : ∀ i, Ω i => ∏ i, w i (ω i)) (fun ω => σ*Equiv.permCongr (finCongr h) (perm n (fun i => g i (ω i)))*σ⁻¹) lemma ambientAverage_factor (h : (∑ i, n i)=l) (σ : Equiv.Perm (Fin l)) {Ω : Fin r → Type*} [∀ i, Fintype (Ω i)] (w : ∀ i, Ω i → ℂ) (g : ∀ i, Ω i → Equiv.Perm (Fin (n i))) : ambientAverage (p := p) n h σ w g= matrix σ*lengthMatrix h (groupedAverage n w g)*(matrix σ).conjTranspose := by subst l simp only [ambientAverage,averageMatrix,groupedAverage,lengthMatrix, Matrix.mul_sum,Matrix.sum_mul,Matrix.mul_smul,Matrix.smul_mul] apply Finset.sum_congr rfl intro ω _ congr 1 change matrix (σ*Equiv.permCongr (finCongr rfl) (perm n (fun i => g i (ω i)))*σ⁻¹)=_ simp only [finCongr_refl,matrix_mul,matrix_inv] rfl lemma matrix_unitary_reverse (σ : Equiv.Perm (Fin l)) : matrix (p := p) σ*(matrix σ).conjTranspose=1 := by rw [← matrix_inv,← matrix_mul,mul_inv_cancel,matrix_one] lemma conjugate_mul {X : Type*} [Fintype X] [DecidableEq X] (U A B : Matrix X X ℂ) (hU : U.conjTranspose*U=1) : (U*A*U.conjTranspose)*(U*B*U.conjTranspose)=U*(A*B)*U.conjTranspose := by calc _ = U*A*(U.conjTranspose*U)*B*U.conjTranspose := by noncomm_ring _ = _ := by rw [hU,mul_one]; noncomm_ring lemma conjugate_pow {X : Type*} [Fintype X] [DecidableEq X] (U A : Matrix X X ℂ) (hU : U.conjTranspose*U=1) (hU' : U*U.conjTranspose=1) (q : ℕ) : (U*A*U.conjTranspose)^q=U*A^q*U.conjTranspose := by induction q with | zero => simpa using hU'.symm | succ q ih => rw [pow_succ,ih,conjugate_mul U _ _ hU,pow_succ] lemma conjugate_moment {X : Type*} [Fintype X] [DecidableEq X] (U A : Matrix X X ℂ) (hU : U.conjTranspose*U=1) (hU' : U*U.conjTranspose=1) (q : ℕ) : Matrix.trace (((U*A*U.conjTranspose).conjTranspose*(U*A*U.conjTranspose))^q)= Matrix.trace ((A.conjTranspose*A)^q) := by have hs : (U*A*U.conjTranspose).conjTranspose=U*A.conjTranspose*U.conjTranspose := by simp only [Matrix.conjTranspose_mul,Matrix.conjTranspose_conjTranspose] noncomm_ring rw [hs,conjugate_mul U _ _ hU,conjugate_pow U _ hU hU'] rw [Matrix.trace_mul_cycle,hU,one_mul] /- The child moment formula survives the literal slot chart, with actual Hom multiplicities. -/ theorem ambient_projected_moment (h : (∑ i, n i)=l) (σ : Equiv.Perm (Fin l)) {Ω : Fin r → Type*} [∀ i, Fintype (Ω i)] (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) (w : ∀ i, Ω i → ℂ) (g : ∀ i, Ω i → Equiv.Perm (Fin (n i))) {q : ℕ} (hq : q ≠ 0) : Matrix.trace (((ambientProjector (p := p) n h σ ρ*ambientAverage n h σ w g).conjTranspose* (ambientProjector n h σ ρ*ambientAverage n h σ w g))^q)= ∏ i, (Module.finrank ℂ ((ρ i).asRepresentation.IntertwiningMap (matrixRepresentation (action p (n i)))) : ℂ)* Matrix.trace (((averageMatrix (ρ i).matrix (w i) (g i)).conjTranspose* averageMatrix (ρ i).matrix (w i) (g i))^q) := by subst l rw [ambientAverage_factor] unfold ambientProjector lengthMatrix rw [conjugate_mul _ _ _ (matrix_unitary σ),conjugate_moment _ _ (matrix_unitary σ) (matrix_unitary_reverse σ)] exact grouped_projected_moment n ρ w g hq end CoordinateSweeps.SignedTensor /-! All definitions are finite and use ordinary probability and trace normalizations. -/ noncomputable section open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder attribute [local instance] Classical.propDecidable noncomputable section attribute [local instance] Classical.propDecidable namespace CoordinateSweeps.FiberBlocks open OrderedBlocks variable {L : Type*} [Fintype L] {r : ℕ} (f : L → Fin r) def size (i : Fin r) : ℕ := Fintype.card {x : L // f x=i} def cards (i : Fin r) : Fin (size f i) ≃ {x : L // f x=i} := (Fintype.equivFin _).symm /- Contiguous finite label fibers give a genuine ordered slot chart. -/ def slots : Fin (∑ i, size f i) ≃ L := (positions (size f)).symm.trans ((Equiv.sigmaCongrRight (cards f)).trans (Equiv.sigmaFiberEquiv f)) lemma total : ∑ i, size f i=Fintype.card L := by simpa only [Fintype.card_fin] using Fintype.card_congr (slots f) lemma label_slots (k : Fin (∑ i, size f i)) : f (slots f k)=((positions (size f)).symm k).1 := by exact (cards f ((positions (size f)).symm k).1 ((positions (size f)).symm k).2).property /- The same chart with the literal cardinal as ambient length. -/ def chart : Fin (Fintype.card L) ≃ L := (finCongr (total f).symm).trans (slots f) /- Permutation changing the contiguous fiber chart to any given ambient chart. -/ def regroup (e : Fin (Fintype.card L) ≃ L) : Equiv.Perm (Fin (Fintype.card L)) := (chart f).trans e.symm lemma label_regroup (e : Fin (Fintype.card L) ≃ L) (k : Fin (Fintype.card L)) : ((positions (size f)).symm ((finCongr (total f)).symm ((regroup f e)⁻¹ k))).1=f (e k) := by have he : slots f ((finCongr (total f)).symm ((regroup f e)⁻¹ k))=e k := by change chart f ((regroup f e).symm k)=e k simp [regroup] rw [← label_slots f,he] end CoordinateSweeps.FiberBlocks namespace CoordinateSweeps.FiberBlocks open OrderedBlocks variable {L : Type*} [Fintype L] {r : ℕ} (f : L → Fin r) lemma slots_positions (i : Fin r) (j : Fin (size f i)) : slots f (positions (size f) ⟨i,j⟩)=(cards f i j).val := by simp [slots] /- Independent permutations of the actual fibers, on the literal ambient set. -/ def action : (∀ i, Equiv.Perm (Fin (size f i))) →* Equiv.Perm L := (slots f).permCongrHom.toMonoidHom.comp (permHom (size f)) lemma action_apply (σ : ∀ i, Equiv.Perm (Fin (size f i))) (i : Fin r) (j : Fin (size f i)) : action f σ (cards f i j).val=(cards f i (σ i j)).val := by rw [← slots_positions f i j,← slots_positions f i (σ i j)] change slots f (perm (size f) σ ((slots f).symm (slots f (positions (size f) ⟨i,j⟩))))=_ rw [Equiv.symm_apply_apply,perm_apply] lemma ambient_action (e : Fin (Fintype.card L) ≃ L) (σ : ∀ i, Equiv.Perm (Fin (size f i))) : Equiv.permCongr e.symm (action f σ)= regroup f e*Equiv.permCongr (finCongr (total f)) (perm (size f) σ)*(regroup f e)⁻¹ := by apply Equiv.ext intro x change e.symm (slots f (perm (size f) σ ((slots f).symm (e x))))=_ simp only [Equiv.Perm.mul_apply,Equiv.permCongr_apply,regroup,chart, Equiv.trans_apply,Equiv.symm_trans_apply,Equiv.Perm.coe_inv] simp /- A label-preserving permutation has genuine induced local permutations. -/ def restrict (σ : Equiv.Perm L) (hσ : ∀ x, f (σ x)=f x) (i : Fin r) : Equiv.Perm (Fin (size f i)) := Equiv.permCongr (cards f i).symm (Equiv.subtypeEquiv σ (by intro x change f x=i ↔ f (σ x)=i rw [hσ x])) lemma restrict_apply (σ : Equiv.Perm L) (hσ : ∀ x, f (σ x)=f x) (i : Fin r) (j : Fin (size f i)) : (cards f i (restrict f σ hσ i j)).val=σ (cards f i j).val := by simp [restrict,Equiv.permCongr_apply] lemma action_restrict (σ : Equiv.Perm L) (hσ : ∀ x, f (σ x)=f x) : action f (restrict f σ hσ)=σ := by apply Equiv.ext intro x let i := f x let j := (cards f i).symm ⟨x,rfl⟩ have hx : (cards f i j).val=x := congrArg Subtype.val ((cards f i).apply_symm_apply ⟨x,rfl⟩) rw [← hx,action_apply,restrict_apply] end CoordinateSweeps.FiberBlocks namespace CoordinateSweeps.SignedTensor open UnitaryIrrep variable {p r l : ℕ} (n : Fin r → ℕ) lemma groupedProjector_commute_average {Ω : Fin r → Type*} [∀ i, Fintype (Ω i)] (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) (w : ∀ i, Ω i → ℂ) (g : ∀ i, Ω i → Equiv.Perm (Fin (n i))) : groupedProjector (p := p) n ρ*groupedAverage n w g=groupedAverage n w g*groupedProjector n ρ := by rw [groupedProjector_factor,groupedAverage_factor] change Matrix.reindexAlgEquiv ℂ ℂ (groupWord n p).symm _* Matrix.reindexAlgEquiv ℂ ℂ (groupWord n p).symm _= Matrix.reindexAlgEquiv ℂ ℂ (groupWord n p).symm _* Matrix.reindexAlgEquiv ℂ ℂ (groupWord n p).symm _ rw [← map_mul,← map_mul,← DependentTensor.tensor_mul,← DependentTensor.tensor_mul] congr 1 apply congrArg DependentTensor.tensor funext i simp only [averageMatrix,Matrix.mul_sum,Matrix.sum_mul,Matrix.mul_smul,Matrix.smul_mul] apply Finset.sum_congr rfl intro ω _ exact congrArg (fun A => w i ω • A) ((ρ i).projector_commute (action p (n i)) (g i ω)).symm lemma ambientProjector_commute_average (h : (∑ i, n i)=l) (σ : Equiv.Perm (Fin l)) {Ω : Fin r → Type*} [∀ i, Fintype (Ω i)] (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) (w : ∀ i, Ω i → ℂ) (g : ∀ i, Ω i → Equiv.Perm (Fin (n i))) : ambientProjector (p := p) n h σ ρ*ambientAverage n h σ w g= ambientAverage n h σ w g*ambientProjector n h σ ρ := by subst l rw [ambientAverage_factor] unfold ambientProjector lengthMatrix rw [conjugate_mul _ _ _ (matrix_unitary σ),conjugate_mul _ _ _ (matrix_unitary σ)] rw [groupedProjector_commute_average] end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor open UnitaryIrrep variable {p r : ℕ} (n : Fin r → ℕ) lemma child_product_bound (s : ℕ) (hn : ∀ i, n i ≤ s) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (n i)))) (T E : Fin r → ℝ) (hT : ∀ i, 0 ≤ T i) (hE : ∀ i, T i ≤ Real.exp (E i)) : (∏ i, (Module.finrank ℂ ((ρ i).asRepresentation.IntertwiningMap (matrixRepresentation (action p (n i)))) : ℝ)*T i) ≤ ((s+1 : ℕ) : ℝ)^(r*((2*p)*(2*p)))*Real.exp (∑ i, E i) := by rw [Finset.prod_mul_distrib] have hm : (∏ i, (Module.finrank ℂ ((ρ i).asRepresentation.IntertwiningMap (matrixRepresentation (action p (n i)))) : ℝ)) ≤ ((s+1 : ℕ) : ℝ)^(r*((2*p)*(2*p))) := by calc _ ≤ ∏ _ : Fin r, ((s+1 : ℕ) : ℝ)^((2*p)*(2*p)) := Finset.prod_le_prod₀ (fun _ _ => Nat.cast_nonneg _) (fun i _ => by exact_mod_cast (multiplicity_le p (n i) (ρ i)).trans (Nat.pow_le_pow_left (Nat.add_le_add_right (hn i) 1) _)) _ = _ := by rw [Finset.prod_const,Finset.card_univ,Fintype.card_fin,← pow_mul,Nat.mul_comm] have ht : (∏ i, T i) ≤ Real.exp (∑ i, E i) := by rw [Real.exp_sum] exact Finset.prod_le_prod₀ (fun i _ => hT i) (fun i _ => hE i) exact mul_le_mul hm ht (Finset.prod_nonneg (fun i _ => hT i)) (by positivity) end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor variable {p r : ℕ} {L : Type*} [Fintype L] (f : L → Fin r) def fiberProjector (e : Fin (Fintype.card L) ≃ L) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (FiberBlocks.size f i)))) := ambientProjector (p := p) (FiberBlocks.size f) (FiberBlocks.total f) (FiberBlocks.regroup f e) ρ def fiberProjectorAlgebra (e : Fin (Fintype.card L) ≃ L) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (FiberBlocks.size f i)))) := ambientProjectorAlgebra (FiberBlocks.size f) (FiberBlocks.total f) (FiberBlocks.regroup f e) ρ lemma fiberProjectorAlgebra_lift (e : Fin (Fintype.card L) ≃ L) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (FiberBlocks.size f i)))) : MonoidAlgebra.lift ℂ _ _ (action p (Fintype.card L)) (fiberProjectorAlgebra f e ρ)= fiberProjector (p := p) f e ρ := ambientProjectorAlgebra_lift _ _ _ _ lemma fiberProjector_posSemidef (e : Fin (Fintype.card L) ≃ L) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (FiberBlocks.size f i)))) : (fiberProjector (p := p) f e ρ).PosSemidef := ambientProjector_posSemidef _ _ _ _ lemma fiberProjector_idempotent (e : Fin (Fintype.card L) ≃ L) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (FiberBlocks.size f i)))) : fiberProjector (p := p) f e ρ*fiberProjector f e ρ=fiberProjector f e ρ := ambientProjector_idempotent _ _ _ _ /- Exact regrouping connects local isotypic products to the literal site labels. -/ theorem finite_fiber_density_domination (e : Fin (Fintype.card L) ≃ L) (s : ℕ) (hp : 0 < p) (hps : p ≤ s) (hn : ∀ i, FiberBlocks.size f i ≤ s) (ρ : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (FiberBlocks.size f i)))) : ∃ (K : Type) (_ : Fintype K) (_ : Nonempty K) (a : K → Fin r → EvenDensity p), (((∏ i, ((ρ i).dimension : ℂ))*((s+1 : ℕ) : ℂ)^(r*(41*p^2))) • ((Fintype.card K : ℂ)⁻¹ • ∑ x, TensorBoard.tensor (fun k => (a x (f (e k))).val))-fiberProjector (p := p) f e ρ).PosSemidef := by obtain ⟨K,hK,nK,a,ha⟩ := finite_ambient_density_domination (FiberBlocks.size f) s hp hps hn ρ (FiberBlocks.total f) (FiberBlocks.regroup f e) let := hK let := nK refine ⟨K,hK,nK,a,?_⟩ simpa only [groupedSite,FiberBlocks.label_regroup,fiberProjector] using ha end CoordinateSweeps.SignedTensor end end end end end end end end open scoped Matrix.Norms.L2Operator end OAI