import Mathlib namespace OAI noncomputable section open scoped BigOperators ComplexConjugate InnerProductSpace namespace SpinAngle.Specht variable {G : Type*} [contextInstance10_21 : Group G] variable {E : Type*} [contextInstance11_21 : NormedAddCommGroup E] [contextInstance11_44 : InnerProductSpace ℂ E] def cyclic (ρ : Representation ℂ G E) (e : E) : Subrepresentation ρ where toSubmodule := Submodule.span ℂ (Set.range fun g => ρ g e) apply_mem_toSubmodule g := by intro v hv induction hv using Submodule.span_induction with | mem v hv => obtain ⟨h, rfl⟩ := hv exact Submodule.subset_span ⟨g * h, by simp⟩ | zero => simp | add x y hx hy ihx ihy => simpa using (Submodule.span ℂ (Set.range fun h => ρ h e)).add_mem ihx ihy | smul a x hx ih => simpa using (Submodule.span ℂ (Set.range fun h => ρ h e)).smul_mem a ih lemma self_mem_cyclic (ρ : Representation ℂ G E) (e : E) : e ∈ cyclic ρ e := by exact Submodule.subset_span ⟨1, by simp⟩ variable [contextInstance32_9 : FiniteDimensional ℂ E] lemma starProjection_rep (ρ : Representation ℂ G E) (hunit : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (U : Subrepresentation ρ) (g : G) (v : E) : U.toSubmodule.starProjection (ρ g v) = ρ g (U.toSubmodule.starProjection v) := by apply U.toSubmodule.eq_starProjection_of_mem_of_inner_eq_zero · exact U.apply_mem_toSubmodule g (U.toSubmodule.starProjection_apply_mem v) · intro w hw have hw' := U.apply_mem_toSubmodule g⁻¹ hw have he : ρ g (ρ g⁻¹ w) = w := by rw [← Module.End.mul_apply, ← map_mul, mul_inv_cancel, map_one, Module.End.one_apply] rw [← map_sub] conv_lhs => rhs; rw [← he] rw [hunit] exact U.toSubmodule.starProjection_inner_eq_zero v _ hw' theorem simple_eigen_submodule_alternative (ρ : Representation ℂ G E) (e : E) (hunit : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (g₀ : G) (c : ℂ) (heig : ρ g₀ e = c • e) (hsimple : ∀ v : E, ρ g₀ v = c • v → v ∈ Submodule.span ℂ {e}) (U : Subrepresentation ρ) : cyclic ρ e ≤ U ∨ ∀ u ∈ U, ∀ v ∈ cyclic ρ e, ⟪v, u⟫_ℂ = 0 := by classical by_cases he : e ∈ U · left exact Submodule.span_le.mpr (by rintro _ ⟨g, rfl⟩; exact U.apply_mem_toSubmodule g he) · right have hp : U.toSubmodule.starProjection e = 0 := by have hpeig : ρ g₀ (U.toSubmodule.starProjection e) = c • U.toSubmodule.starProjection e := by rw [← starProjection_rep ρ hunit U g₀ e, heig, map_smul] obtain ⟨a, ha⟩ := Submodule.mem_span_singleton.mp (hsimple _ hpeig) by_cases ha0 : a = 0 · simpa [ha0] using ha.symm · apply False.elim apply he have hm := U.toSubmodule.starProjection_apply_mem e rw [← ha] at hm have hh := U.toSubmodule.smul_mem a⁻¹ hm change e ∈ U.toSubmodule simpa [smul_smul, ha0] using hh have heorth (u : E) (hu : u ∈ U) : ⟪e, u⟫_ℂ = 0 := by have hh := U.toSubmodule.starProjection_inner_eq_zero e u hu simpa [hp] using hh intro u hu v hv have horth (g : G) : ⟪ρ g e, u⟫_ℂ = 0 := by have hgu : ρ g⁻¹ u ∈ U := U.apply_mem_toSubmodule g⁻¹ hu have hi := hunit g e (ρ g⁻¹ u) simpa only [← Module.End.mul_apply, ← map_mul, mul_inv_cancel, map_one, Module.End.one_apply] using hi.trans (heorth _ hgu) induction hv using Submodule.span_induction with | mem v hv => obtain ⟨g, rfl⟩ := hv; exact horth g | zero => simp | add x y hx hy ihx ihy => simp [ihx, ihy] | smul a x hx ih => simp [ih] theorem cyclic_irreducible_of_simple_eigen (ρ : Representation ℂ G E) (e : E) (he : e ≠ 0) (hunit : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (g₀ : G) (c : ℂ) (heig : ρ g₀ e = c • e) (hsimple : ∀ v : E, ρ g₀ v = c • v → v ∈ Submodule.span ℂ {e}) : Representation.IsIrreducible (cyclic ρ e).toRepresentation := by let S := cyclic ρ e let eS : S.toSubmodule := ⟨e, self_mem_cyclic ρ e⟩ have heS : eS ≠ 0 := by intro hh; apply he; exact congrArg Subtype.val hh have htb : (⊤ : Subrepresentation S.toRepresentation) ≠ ⊥ := by intro hh have hel : eS ∈ (⊤ : Subrepresentation S.toRepresentation) := by trivial rw [hh] at hel exact heS hel let : Nontrivial (Subrepresentation S.toRepresentation) := ⟨⟨⊤, ⊥, htb⟩⟩ refine ⟨fun U => ?_⟩ let U' : Subrepresentation ρ := { toSubmodule := U.toSubmodule.map S.toSubmodule.subtype apply_mem_toSubmodule := by intro g v hv obtain ⟨u, hu, rfl⟩ := hv exact ⟨S.toRepresentation g u, U.apply_mem_toSubmodule g hu, rfl⟩ } rcases simple_eigen_submodule_alternative ρ e hunit g₀ c heig hsimple U' with h | h · right apply top_unique intro u _ obtain ⟨v, hv, heq⟩ := h u.property have hvu : v = u := Subtype.ext heq exact hvu ▸ hv · left apply bot_unique intro u hu have hum : (u : E) ∈ U' := ⟨u, hu, rfl⟩ have hh := h _ hum _ u.property have hz : (u : E) = 0 := inner_self_eq_zero.mp hh exact Subtype.ext hz end SpinAngle.Specht noncomputable section open scoped BigOperators ComplexConjugate InnerProductSpace Matrix namespace SpinAngle.TensorRep variable {ι n : Type*} [contextInstance135_23 : Fintype ι] [contextInstance135_35 : DecidableEq ι] [contextInstance135_51 : Fintype n] [contextInstance135_63 : DecidableEq n] def tensor (a : ι → Matrix n n ℂ) : Matrix (ι → n) (ι → n) ℂ := fun x y => ∏ i, a i (x i) (y i) lemma tensor_star {ι : Type*} {n : Type*} [contextInstance135_23 : Fintype ι] [DecidableEq ι] [Fintype n] [DecidableEq n] (a : ι → Matrix n n ℂ) : tensor (fun i => star (a i)) = star (tensor a) := by ext x y simp [tensor, Matrix.star_apply] lemma tensor_mul {ι : Type*} {n : Type*} [contextInstance135_23 : Fintype ι] [contextInstance135_35 : DecidableEq ι] [contextInstance135_51 : Fintype n] [DecidableEq n] (a b : ι → Matrix n n ℂ) : tensor (fun i => a i * b i) = tensor a * tensor b := by ext x y simp only [tensor, Matrix.mul_apply, ← Finset.prod_mul_distrib] symm simpa using (Finset.sum_prod_piFinset (ι := ι) Finset.univ (fun i j => a i (x i) j * b i j (y i))) lemma tensor_one {ι : Type*} {n : Type*} [contextInstance135_23 : Fintype ι] [DecidableEq ι] [Fintype n] [contextInstance135_63 : DecidableEq n] : tensor (fun _ : ι => (1 : Matrix n n ℂ)) = 1 := by ext x y classical by_cases h : x = y · subst y simp [tensor] · obtain ⟨i, hi⟩ := Function.ne_iff.mp h simp only [tensor, Matrix.one_apply, ite_eq_right h] apply Finset.prod_eq_zero (Finset.mem_univ i) simp [hi] lemma tensor_diagonal {ι : Type*} {n : Type*} [contextInstance135_23 : Fintype ι] [DecidableEq ι] [Fintype n] [contextInstance135_63 : DecidableEq n] (a : ι → n → ℂ) : tensor (fun i => Matrix.diagonal (a i)) = Matrix.diagonal (fun x => ∏ i, a i (x i)) := by ext x y classical by_cases h : x = y · subst y simp [tensor] · obtain ⟨i, hi⟩ := Function.ne_iff.mp h simp only [tensor, Matrix.diagonal_apply, ite_eq_right h] apply Finset.prod_eq_zero (Finset.mem_univ i) simp [hi] lemma toEuclideanLin_mul {ι : Type*} {n : Type*} [Fintype ι] [DecidableEq ι] [contextInstance135_51 : Fintype n] [contextInstance135_63 : DecidableEq n] (a b : Matrix n n ℂ) : Matrix.toEuclideanLin (a * b) = Matrix.toEuclideanLin a * Matrix.toEuclideanLin b := by ext v x change ((a * b) *ᵥ (WithLp.ofLp v)) x = (a *ᵥ (b *ᵥ (WithLp.ofLp v))) x rw [Matrix.mulVec_mulVec] lemma toEuclideanLin_one {ι : Type*} {n : Type*} [Fintype ι] [DecidableEq ι] [contextInstance135_51 : Fintype n] [contextInstance135_63 : DecidableEq n] : Matrix.toEuclideanLin (1 : Matrix n n ℂ) = 1 := by ext v x change ((1 : Matrix n n ℂ) *ᵥ (WithLp.ofLp v)) x = v x simp def action : Matrix n n ℂ →* Module.End ℂ (EuclideanSpace ℂ (ι → n)) where toFun a := Matrix.toEuclideanLin (tensor (fun _ : ι => a)) map_one' := by rw [tensor_one, toEuclideanLin_one (ι := ι)] map_mul' a b := by rw [tensor_mul, toEuclideanLin_mul (ι := ι)] def rep : Representation ℂ (Matrix.unitaryGroup n ℂ) (EuclideanSpace ℂ (ι → n)) := action.comp (Matrix.unitaryGroup n ℂ).subtype lemma tensor_mem_unitary (a : ι → Matrix.unitaryGroup n ℂ) : tensor (fun i => (a i : Matrix n n ℂ)) ∈ Matrix.unitaryGroup (ι → n) ℂ := by rw [Matrix.mem_unitaryGroup_iff', ← tensor_star, ← tensor_mul] convert tensor_one (ι := ι) (n := n) using 1 congr 1 funext i exact (a i).property.1 lemma toEuclideanLin_unitary (a : Matrix n n ℂ) (ha : a ∈ Matrix.unitaryGroup n ℂ) (v w : EuclideanSpace ℂ n) : ⟪Matrix.toEuclideanLin a v, Matrix.toEuclideanLin a w⟫_ℂ = ⟪v, w⟫_ℂ := by rw [← LinearMap.adjoint_inner_right] rw [← Matrix.toEuclideanLin_conjTranspose_eq_adjoint] change ⟪v, (Matrix.toEuclideanLin (star a) * Matrix.toEuclideanLin a) w⟫_ℂ = _ rw [← toEuclideanLin_mul (ι := n), ha.1, toEuclideanLin_one (ι := n)] rfl lemma rep_unitary (g : Matrix.unitaryGroup n ℂ) (v w : EuclideanSpace ℂ (ι → n)) : ⟪rep g v, rep g w⟫_ℂ = ⟪v, w⟫_ℂ := toEuclideanLin_unitary _ (tensor_mem_unitary (fun _ : ι => g)) v w lemma tensor_reindex {ι : Type*} {n : Type*} [contextInstance135_23 : Fintype ι] [DecidableEq ι] [Fintype n] [DecidableEq n] (a : Matrix n n ℂ) (σ : Equiv.Perm ι) (x y : ι → n) : tensor (fun _ : ι => a) (x ∘ σ) (y ∘ σ) = tensor (fun _ : ι => a) x y := by exact Equiv.prod_comp σ (fun i => a (x i) (y i)) lemma action_diagonal (a : n → ℂ) (v : EuclideanSpace ℂ (ι → n)) (x : ι → n) : action (Matrix.diagonal a) v x = (∏ i, a (x i)) * v x := by change (Matrix.toEuclideanLin (tensor (fun _ : ι => Matrix.diagonal a)) v) x = _ rw [tensor_diagonal] change (Matrix.diagonal (fun x : ι → n => ∏ i, a (x i)) *ᵥ WithLp.ofLp v) x = _ exact Matrix.mulVec_diagonal _ _ _ end SpinAngle.TensorRep namespace SpinAngle.Specht open scoped Classical variable {G : Type*} [contextInstance243_21 : Group G] namespace PermutationSpace open scoped Classical variable {X : Type*} [contextInstance247_21 : Fintype X] [contextInstance247_33 : MulAction G X] def rep : Representation ℂ G (EuclideanSpace ℂ X) where toFun g := { toFun v := WithLp.toLp 2 (fun x => v (g⁻¹ • x)) map_add' := by intro v w; rfl map_smul' := by intro a v; rfl } map_one' := by ext v x; simp map_mul' := by intro g h ext v x change v ((g * h)⁻¹ • x) = v (h⁻¹ • g⁻¹ • x) simp [mul_smul] omit [Fintype X] in @[simp] lemma rep_apply (g : G) (v : EuclideanSpace ℂ X) (x : X) : rep g v x = v (g⁻¹ • x) := rfl lemma rep_unitary (g : G) (v w : EuclideanSpace ℂ X) : ⟪rep g v, rep g w⟫_ℂ = ⟪v, w⟫_ℂ := by simp only [PiLp.inner_apply, rep_apply] exact Equiv.sum_comp (MulAction.toPerm g⁻¹) (fun x => inner ℂ (v x) (w x)) omit [Fintype X] in lemma rep_single (g : G) (x : X) : rep g (EuclideanSpace.single x (1 : ℂ)) = EuclideanSpace.single (g • x) (1 : ℂ) := by classical ext y simp only [rep_apply, PiLp.single_apply, inv_smul_eq_iff] variable {H : Type*} [contextInstance278_21 : Group H] [contextInstance278_31 : Fintype H] def alternator (ι : H →* G) (χ : H →* ℂ) : Module.End ℂ (EuclideanSpace ℂ X) := ∑ h, χ h • rep (ι h) omit [Fintype X] in lemma alternator_apply (ι : H →* G) (χ : H →* ℂ) (v : EuclideanSpace ℂ X) : alternator ι χ v = ∑ h, χ h • rep (ι h) v := by simp [alternator, LinearMap.sum_apply] omit [Fintype X] in lemma alternator_stable (ι : H →* G) (χ : H →* ℂ) (U : Subrepresentation (rep : Representation ℂ G (EuclideanSpace ℂ X))) (v : EuclideanSpace ℂ X) (hv : v ∈ U) : alternator ι χ v ∈ U := by rw [alternator_apply] exact U.toSubmodule.sum_mem fun h _ => U.toSubmodule.smul_mem _ (U.apply_mem_toSubmodule _ hv) omit [Fintype X] in lemma alternator_translate (ι : H →* G) (χ : H →* ℂ) (a : H) (v : EuclideanSpace ℂ X) : alternator ι χ (rep (ι a) v) = χ a⁻¹ • alternator ι χ v := by rw [alternator_apply, alternator_apply, Finset.smul_sum] have he := Equiv.sum_comp (Equiv.mulRight a) (fun h => χ a⁻¹ • (χ h • (rep (ι h) v : EuclideanSpace ℂ X))) rw [← he] apply Finset.sum_congr rfl intro h _ simp only [Equiv.coe_mulRight, smul_smul, map_mul] rw [show χ a⁻¹ * (χ h * χ a) = χ h by calc _ = χ h * (χ a⁻¹ * χ a) := by ring _ = χ h := by rw [← map_mul, inv_mul_cancel, map_one, mul_one]] simp only [Module.End.mul_apply] omit [Fintype X] in lemma alternator_single_zero (ι : H →* G) (χ : H →* ℂ) (x : X) (a : H) (ha : ι a • x = x) (hs : χ a = -1) : alternator ι χ (EuclideanSpace.single x (1 : ℂ)) = 0 := by have hh := alternator_translate ι χ a (EuclideanSpace.single x (1 : ℂ)) rw [rep_single, ha, map_inv, hs, inv_neg, inv_one, neg_smul, one_smul] at hh have hz := congrArg (fun z => z + alternator ι χ (EuclideanSpace.single x (1 : ℂ))) hh have ht : (2 : ℂ) • alternator ι χ (EuclideanSpace.single x (1 : ℂ)) = 0 := by simpa [two_smul] using hz exact (smul_eq_zero.mp ht).resolve_left (by norm_num) omit [Fintype X] in lemma alternator_coeff (ι : H →* G) (χ : H →* ℂ) (v : EuclideanSpace ℂ X) (x : X) : alternator ι χ v x = ∑ h, χ h * v ((ι h)⁻¹ • x) := by simp [alternator_apply, WithLp.ofLp_sum, Finset.sum_apply] omit [Fintype X] in lemma polytabloid_base [DecidableEq X] (ι : H →* G) (χ : H →* ℂ) (b : X) (hfree : ∀ h : H, ι h • b = b → h = 1) : alternator ι χ (EuclideanSpace.single b (1 : ℂ)) b = 1 := by classical rw [alternator_coeff, Finset.sum_eq_single 1] · simp · intro h _ hh have hn : (ι h)⁻¹ • b ≠ b := by intro he have hi : h⁻¹ = 1 := hfree h⁻¹ (by simpa using he) exact hh (inv_eq_one.mp hi) simp [hn] · simp lemma polytabloid_inner (ι : H →* G) (χ : H →* ℂ) (b : X) (hreal : ∀ h, conj (χ h) = χ h) (hinv : ∀ h, χ h⁻¹ = χ h) (v : EuclideanSpace ℂ X) : ⟪alternator ι χ (EuclideanSpace.single b (1 : ℂ)), v⟫_ℂ = alternator ι χ v b := by classical rw [alternator_apply, sum_inner, alternator_coeff] simp only [inner_smul_left, rep_single, EuclideanSpace.inner_single_left, map_one, one_mul, hreal] have hh := Equiv.sum_comp (Equiv.inv H) (fun h => χ h * v ((ι h)⁻¹ • b)) simpa only [Equiv.inv_apply, map_inv, inv_inv, hinv] using hh theorem alternator_rank_one (ι : H →* G) (χ : H →* ℂ) (b : X) (hreal : ∀ h, conj (χ h) = χ h) (hinv : ∀ h, χ h⁻¹ = χ h) (hfree : ∀ h : H, ι h • b = b → h = 1) (halt : ∀ x : X, (∃ a : H, ι a • x = x ∧ χ a = -1) ∨ ∃ a : H, x = ι a • b) (v : EuclideanSpace ℂ X) : alternator ι χ v = ⟪alternator ι χ (EuclideanSpace.single b (1 : ℂ)), v⟫_ℂ • alternator ι χ (EuclideanSpace.single b (1 : ℂ)) := by classical let e := alternator ι χ (EuclideanSpace.single b (1 : ℂ)) have hs (x : X) : alternator ι χ (EuclideanSpace.single x (1 : ℂ)) ∈ Submodule.span ℂ {e} := by rcases halt x with ⟨a, ha, hχ⟩ | ⟨a, rfl⟩ · rw [alternator_single_zero ι χ x a ha hχ] exact Submodule.zero_mem _ · rw [← rep_single, alternator_translate] exact Submodule.smul_mem _ _ (Submodule.subset_span (Set.mem_singleton e)) have hv : alternator ι χ v ∈ Submodule.span ℂ {e} := by have hexp := (EuclideanSpace.basisFun X ℂ).sum_repr v simp only [EuclideanSpace.basisFun_repr, EuclideanSpace.basisFun_apply] at hexp rw [← hexp, map_sum] exact Submodule.sum_mem _ fun x _ => by rw [map_smul] exact Submodule.smul_mem _ _ (hs x) obtain ⟨a, ha⟩ := Submodule.mem_span_singleton.mp hv have heb : e b = 1 := polytabloid_base ι χ b hfree have hab : a = alternator ι χ v b := by have hh := congrArg (fun z : EuclideanSpace ℂ X => z b) ha simpa only [PiLp.smul_apply, smul_eq_mul, heb, mul_one] using hh rw [← ha, hab, polytabloid_inner ι χ b hreal hinv] end PermutationSpace variable {α : Type*} [contextInstance389_21 : Fintype α] def complexSign : Equiv.Perm α →* ℂ where toFun g := ((Equiv.Perm.sign g : ℤˣ) : ℤ) map_one' := by simp map_mul' := by intro g h; simp @[simp] lemma complexSign_conj (g : Equiv.Perm α) : conj (complexSign g) = complexSign g := by simp [complexSign] @[simp] lemma complexSign_inv (g : Equiv.Perm α) : complexSign g⁻¹ = complexSign g := by simp [complexSign] @[simp] lemma complexSign_swap (x y : α) (h : x ≠ y) : complexSign (Equiv.swap x y) = -1 := by simp [complexSign, Equiv.Perm.sign_swap h] end SpinAngle.Specht namespace SpinAngle.Young abbrev Cells (D : YoungDiagram) := {p : ℕ × ℕ // p ∈ D.cells} lemma fin_le_strictMono {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) (i : Fin n) : i.val ≤ f i := by have aux : ∀ k (hk : k < n), k ≤ f ⟨k, hk⟩ := by intro k induction k with | zero => intro hk; exact Nat.zero_le _ | succ k ih => intro hk have hk' : k < n := by omega have hl := hf (show (⟨k, hk'⟩ : Fin n) < ⟨k + 1, hk⟩ from by simp) have hi := ih hk' omega exact aux i.val i.isLt lemma card_column (D : YoungDiagram) (j : ℕ) : (Finset.univ.filter fun x : Cells D => x.val.2 = j).card = D.colLen j := by classical let e : {x : Cells D // x.val.2 = j} ≃ Fin (D.colLen j) := { toFun := fun x => ⟨x.val.val.1, YoungDiagram.mem_iff_lt_colLen.mp (by have hx : (x.val.val.1, x.val.val.2) ∈ D := x.val.property simpa only [x.property] using hx)⟩ invFun := fun i => ⟨⟨(i, j), YoungDiagram.mem_iff_lt_colLen.mpr i.isLt⟩, rfl⟩ left_inv := by intro x; apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.property.symm right_inv := by intro i; rfl } rw [← Fintype.card_subtype] exact (Fintype.card_congr e).trans (Fintype.card_fin _) lemma card_row_subtype (D : YoungDiagram) (i : ℕ) : Fintype.card {x : Cells D // x.val.1 = i} = D.rowLen i := by let e : {x : Cells D // x.val.1 = i} ≃ Fin (D.rowLen i) := { toFun := fun x => ⟨x.val.val.2, YoungDiagram.mem_iff_lt_rowLen.mp (by have hx : (x.val.val.1, x.val.val.2) ∈ D := x.val.property simpa only [x.property] using hx)⟩ invFun := fun j => ⟨⟨(i, j), YoungDiagram.mem_iff_lt_rowLen.mpr j.isLt⟩, rfl⟩ left_inv := by intro x; apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.property.symm rfl right_inv := by intro j; rfl } exact (Fintype.card_congr e).trans (Fintype.card_fin _) lemma eq_of_rowPreserving (D E : YoungDiagram) (e : Cells D ≃ Cells E) (he : ∀ x, (e x).val.1 = x.val.1) : D = E := by have hr (i : ℕ) : D.rowLen i = E.rowLen i := by rw [← card_row_subtype D i, ← card_row_subtype E i] exact Fintype.card_congr (e.subtypeEquiv (fun x => by rw [he])) apply YoungDiagram.ext ext p obtain ⟨i, j⟩ := p change (i, j) ∈ D ↔ (i, j) ∈ E simp only [YoungDiagram.mem_iff_lt_rowLen, hr] theorem column_sort_le (D : YoungDiagram) (r : Cells D → ℕ) (hinj : ∀ x y : Cells D, x.val.2 = y.val.2 → r x = r y → x = y) : ∃ σ : Equiv.Perm (Cells D), (∀ x, (σ x).val.2 = x.val.2) ∧ ∀ x, x.val.1 ≤ r (σ x) := by classical let S (j : ℕ) : Finset ℕ := (Finset.univ.filter fun x : Cells D => x.val.2 = j).image r have hc (j : ℕ) : (S j).card = D.colLen j := by change ((Finset.univ.filter fun x : Cells D => x.val.2 = j).image r).card = _ rw [Finset.card_image_iff.mpr (show Set.InjOn r ↑(Finset.univ.filter fun x : Cells D => x.val.2 = j) from by intro x hx y hy hxy exact hinj x y ((Finset.mem_filter.mp hx).2.trans (Finset.mem_filter.mp hy).2.symm) hxy)] exact card_column D j let F (j : ℕ) := (S j).orderEmbOfFin (hc j) have hpre (x : Cells D) : ∃ y : Cells D, y.val.2 = x.val.2 ∧ r y = F x.val.2 ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩ := by have hm := (S x.val.2).orderEmbOfFin_mem (hc _) ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩ obtain ⟨y, hy, he⟩ := Finset.mem_image.mp hm exact ⟨y, (Finset.mem_filter.mp hy).2, he⟩ choose f hcol hr using hpre have hf : Function.Injective f := by intro x y he have hj : x.val.2 = y.val.2 := (hcol x).symm.trans ((congrArg (fun z : Cells D => z.val.2) he).trans (hcol y)) have hi : x.val.1 = y.val.1 := by have hval : F x.val.2 ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩ = F y.val.2 ⟨y.val.1, YoungDiagram.mem_iff_lt_colLen.mp y.property⟩ := by rw [← hr x, ← hr y, he] have hy : y.val.1 < D.colLen x.val.2 := by rw [hj]; exact YoungDiagram.mem_iff_lt_colLen.mp y.property have hid : F y.val.2 ⟨y.val.1, YoungDiagram.mem_iff_lt_colLen.mp y.property⟩ = F x.val.2 ⟨y.val.1, hy⟩ := by have aux (j k i : ℕ) (hk : j = k) (hi : i < D.colLen j) (hi' : i < D.colLen k) : F j ⟨i, hi⟩ = F k ⟨i, hi'⟩ := by subst k; rfl exact aux _ _ _ hj.symm _ _ rw [hid] at hval exact congrArg Fin.val ((F x.val.2).injective hval) exact Subtype.ext (Prod.ext hi hj) let σ : Equiv.Perm (Cells D) := Equiv.ofBijective f ⟨hf, Finite.surjective_of_injective hf⟩ refine ⟨σ, hcol, ?_⟩ have hle (x : Cells D) : x.val.1 ≤ r (σ x) := by change x.val.1 ≤ r (f x) rw [hr] exact fin_le_strictMono _ (F x.val.2).strictMono ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩ exact hle lemma rowRankSum_le (D : YoungDiagram) (r : Cells D → ℕ) (hinj : ∀ x y : Cells D, x.val.2 = y.val.2 → r x = r y → x = y) : ∑ x : Cells D, x.val.1 ≤ ∑ x : Cells D, r x := by obtain ⟨σ, -, hσ⟩ := column_sort_le D r hinj rw [← Equiv.sum_comp σ r] exact Finset.sum_le_sum fun x _ => hσ x theorem column_sort (D : YoungDiagram) (r : Cells D → ℕ) (hinj : ∀ x y : Cells D, x.val.2 = y.val.2 → r x = r y → x = y) (hsum : ∑ x : Cells D, r x = ∑ x : Cells D, x.val.1) : ∃ σ : Equiv.Perm (Cells D), (∀ x, (σ x).val.2 = x.val.2) ∧ ∀ x, r (σ x) = x.val.1 := by obtain ⟨σ, hc, hle⟩ := column_sort_le D r hinj refine ⟨σ, hc, ?_⟩ have heq : ∑ x : Cells D, x.val.1 = ∑ x : Cells D, r (σ x) := by rw [Equiv.sum_comp σ r, hsum] have hh := (Finset.sum_eq_sum_iff_of_le (s := Finset.univ) (fun x _ => hle x)).mp heq intro x exact (hh x (Finset.mem_univ x)).symm end SpinAngle.Young namespace SpinAngle.PolynomialCarrier open scoped Classical Matrix open SpinAngle.Specht SpinAngle.Young variable {ι n : Type*} [contextInstance532_23 : Fintype ι] [contextInstance532_35 : DecidableEq ι] [contextInstance532_51 : Fintype n] [contextInstance532_63 : DecidableEq n] instance coloringAction : MulAction (Equiv.Perm ι) (ι → n) where smul g x := x ∘ g.symm one_smul _ := rfl mul_smul _ _ _ := rfl lemma site_apply {ι : Type*} {n : Type*} [Fintype ι] [DecidableEq ι] [Fintype n] [DecidableEq n] (g : Equiv.Perm ι) (v : EuclideanSpace ℂ (ι → n)) (x : ι → n) : PermutationSpace.rep g v x = v (x ∘ g) := rfl def reindex (g : Equiv.Perm ι) : (ι → n) ≃ (ι → n) where toFun x := x ∘ g invFun x := x ∘ g.symm left_inv x := by funext i; simp right_inv x := by funext i; simp lemma action_site_commute (a : Matrix n n ℂ) (g : Equiv.Perm ι) (v : EuclideanSpace ℂ (ι → n)) : TensorRep.action a (PermutationSpace.rep g v) = PermutationSpace.rep g (TensorRep.action a v) := by ext x change (∑ y, TensorRep.tensor (fun _ : ι => a) x y * v (y ∘ g)) = ∑ y, TensorRep.tensor (fun _ : ι => a) (x ∘ g) y * v y rw [← Equiv.sum_comp (reindex (n := n) g) (fun y => TensorRep.tensor (fun _ : ι => a) (x ∘ g) y * v y)] apply Finset.sum_congr rfl intro y _ change _ = TensorRep.tensor (fun _ : ι => a) (x ∘ g) (y ∘ g) * v (y ∘ g) rw [TensorRep.tensor_reindex] def columnGroup (D : YoungDiagram) : Subgroup (Equiv.Perm (Cells D)) where carrier := {g | ∀ x, (g x).val.2 = x.val.2} one_mem' := fun _ => rfl mul_mem' := by intro g h hg hh x; exact (hg (h x)).trans (hh x) inv_mem' := by intro g hg x simpa using (hg (g⁻¹ x)).symm variable (D : YoungDiagram) (q : ℕ) abbrev TensorSpace := EuclideanSpace ℂ (Cells D → Fin q) def alternating : Subrepresentation (TensorRep.rep (ι := Cells D) (n := Fin q)) where toSubmodule := { carrier := {v | ∀ g : columnGroup D, PermutationSpace.rep (g : Equiv.Perm (Cells D)) v = complexSign g.val • v} zero_mem' := by intro g; simp add_mem' := by intro x y hx hy g; simp [map_add, hx g, hy g, smul_add] smul_mem' := by intro c x hx g; simp [map_smul, hx g, smul_comm c] } apply_mem_toSubmodule a := by intro v hv g change PermutationSpace.rep g.val (TensorRep.action (a : Matrix (Fin q) (Fin q) ℂ) v) = _ rw [← action_site_commute, hv g, map_smul] rfl variable (hq : D.colLen 0 ≤ q) def rowColor : Cells D → Fin q := fun x => ⟨x.val.1, by have hm := D.up_left_mem le_rfl (Nat.zero_le x.val.2) x.property have hl := YoungDiagram.mem_iff_lt_colLen.mp hm exact lt_of_lt_of_le hl hq⟩ def highest : TensorSpace D q := by classical exact PermutationSpace.alternator (columnGroup D).subtype (complexSign.comp (columnGroup D).subtype) (EuclideanSpace.single (rowColor D q hq) (1 : ℂ)) lemma rowColor_column_free {ι : Type*} [Fintype ι] [DecidableEq ι] (D : YoungDiagram) (q : ℕ) (hq : D.colLen 0 ≤ q) (g : columnGroup D) (hg : g.val • rowColor D q hq = rowColor D q hq) : g = 1 := by apply Subtype.ext apply Equiv.Perm.ext intro x apply Subtype.ext apply Prod.ext · have hh := congrFun hg (g.val x) have hh' := congrArg Fin.val hh change (g.val x).val.1 = x.val.1 change ((g.val x |> g.val.symm)).val.1 = (g.val x).val.1 at hh' simp only [Equiv.symm_apply_apply] at hh' exact hh'.symm · exact g.property x lemma highest_base : highest D q hq (rowColor D q hq) = 1 := by classical unfold highest exact PermutationSpace.polytabloid_base (columnGroup D).subtype (complexSign.comp (columnGroup D).subtype) (rowColor D q hq) (rowColor_column_free (ι := Cells D) D q hq) lemma highest_ne_zero : highest D q hq ≠ 0 := by intro he have hh := congrArg (fun v : TensorSpace D q => v (rowColor D q hq)) he simp [highest_base] at hh end SpinAngle.PolynomialCarrier namespace SpinAngle.Specht.PermutationSpace open scoped Classical variable {G H X : Type*} [contextInstance635_25 : Group G] [contextInstance635_35 : Group H] [contextInstance635_45 : Fintype H] [contextInstance635_57 : Fintype X] [contextInstance635_69 : MulAction G X] lemma rep_alternator {G : Type*} {H : Type*} {X : Type*} [contextInstance635_25 : Group G] [contextInstance635_35 : Group H] [contextInstance635_45 : Fintype H] [Fintype X] [contextInstance635_69 : MulAction G X] (ι : H →* G) (χ : H →* ℂ) (a : H) (v : EuclideanSpace ℂ X) : rep (ι a) (alternator ι χ v) = χ a⁻¹ • alternator ι χ v := by rw [alternator_apply, map_sum, Finset.smul_sum] have he := Equiv.sum_comp (Equiv.mulLeft a) (fun h => χ a⁻¹ • (χ h • (rep (ι h) v : EuclideanSpace ℂ X))) rw [← he] apply Finset.sum_congr rfl intro h _ simp only [Equiv.coe_mulLeft, smul_smul, map_mul, map_smul] rw [show χ a⁻¹ * (χ a * χ h) = χ h by rw [← mul_assoc, ← map_mul, inv_mul_cancel, map_one, one_mul]] rfl end SpinAngle.Specht.PermutationSpace namespace SpinAngle.PolynomialCarrier open scoped Classical Matrix open SpinAngle.Specht SpinAngle.Young variable (D : YoungDiagram) (q : ℕ) (hq : D.colLen 0 ≤ q) lemma highest_mem : highest D q hq ∈ alternating D q := by classical intro g change PermutationSpace.rep g.val (PermutationSpace.alternator _ _ _) = _ have hh := PermutationSpace.rep_alternator (columnGroup D).subtype (complexSign.comp (columnGroup D).subtype) g (EuclideanSpace.single (rowColor D q hq) (1 : ℂ)) change PermutationSpace.rep g.val (highest D q hq) = complexSign (g.val⁻¹) • highest D q hq at hh rw [complexSign_inv] at hh exact hh lemma alternating_collision (v : TensorSpace D q) (hv : v ∈ alternating D q) (x : Cells D → Fin q) (i j : Cells D) (hij : i ≠ j) (hc : i.val.2 = j.val.2) (hx : x i = x j) : v x = 0 := by classical let : DecidableEq (Cells D) := Classical.decEq _ let g : columnGroup D := ⟨Equiv.swap i j, by intro k by_cases hki : k = i · subst k; simp [hc] by_cases hkj : k = j · subst k; simp [hc] rw [Equiv.swap_apply_of_ne_of_ne hki hkj]⟩ have hperm : x ∘ (g : Equiv.Perm (Cells D)) = x := by funext k change x (Equiv.swap i j k) = x k by_cases hki : k = i · subst k; simp [hx] by_cases hkj : k = j · subst k; simp [hx] rw [Equiv.swap_apply_of_ne_of_ne hki hkj] have hh := congrArg (fun w : TensorSpace D q => w x) (hv g) simp only [site_apply, hperm, PiLp.smul_apply, smul_eq_mul] at hh have hg : complexSign g.val = -1 := by simpa only [g] using complexSign_swap i j hij rw [hg, neg_one_mul] at hh linear_combination (1/2 : ℂ) * hh lemma alternating_injective (v : TensorSpace D q) (hv : v ∈ alternating D q) (x : Cells D → Fin q) (hx : v x ≠ 0) : ∀ i j : Cells D, i.val.2 = j.val.2 → (x i).val = (x j).val → i = j := by intro i j hc he by_contra hij exact hx (alternating_collision D q v hv x i j hij hc (Fin.ext he)) lemma alternating_minimal_weight (v : TensorSpace D q) (hv : v ∈ alternating D q) (x : Cells D → Fin q) (hw : ∑ i, (x i).val = ∑ i : Cells D, i.val.1) : v x = v (rowColor D q hq) * highest D q hq x := by by_cases hx : v x = 0 · by_cases he : highest D q hq x = 0 · rw [hx, he, mul_zero] · obtain ⟨σ, hc, hr⟩ := column_sort D (fun i => (x i).val) (alternating_injective D q (highest D q hq) (highest_mem D q hq) x he) hw let g : columnGroup D := ⟨σ, hc⟩ have hcolor : x ∘ σ = rowColor D q hq := by funext i; exact Fin.ext (hr i) have hh := congrArg (fun w : TensorSpace D q => w x) (hv g) simp only [site_apply, PiLp.smul_apply, smul_eq_mul] at hh change v (x ∘ σ) = complexSign σ * v x at hh rw [hcolor, hx, mul_zero] at hh rw [hx, hh, zero_mul] · obtain ⟨σ, hc, hr⟩ := column_sort D (fun i => (x i).val) (alternating_injective D q v hv x hx) hw let g : columnGroup D := ⟨σ, hc⟩ have hcolor : x ∘ σ = rowColor D q hq := by funext i; exact Fin.ext (hr i) have hh := congrArg (fun w : TensorSpace D q => w x) (hv g) have he := congrArg (fun w : TensorSpace D q => w x) (highest_mem D q hq g) simp only [site_apply, PiLp.smul_apply, smul_eq_mul] at hh he change v (x ∘ σ) = complexSign σ * v x at hh change highest D q hq (x ∘ σ) = complexSign σ * highest D q hq x at he rw [hcolor] at hh he rw [highest_base] at he rw [hh] calc v x = v x * 1 := (mul_one _).symm _ = v x * (complexSign σ * highest D q hq x) := by rw [← he] _ = complexSign σ * v x * highest D q hq x := by ring end SpinAngle.PolynomialCarrier namespace SpinAngle.PolynomialCarrier open scoped Classical Matrix open SpinAngle.Specht SpinAngle.Young variable {ι : Type*} [contextInstance741_21 : Fintype ι] [contextInstance741_33 : DecidableEq ι] {q : ℕ} lemma diagonal_single (a : Fin q → ℂ) (b : ι → Fin q) : TensorRep.action (Matrix.diagonal a) (EuclideanSpace.single b (1 : ℂ)) = (∏ i, a (b i)) • EuclideanSpace.single b (1 : ℂ) := by ext x rw [TensorRep.action_diagonal] simp only [PiLp.smul_apply, smul_eq_mul, PiLp.single_apply] by_cases h : x = b · subst x; rfl · simp [h] lemma diagonal_unitary {z : ℂ} (hz : star z * z = 1) : Matrix.diagonal (fun i : Fin q => z ^ i.val) ∈ Matrix.unitaryGroup (Fin q) ℂ := by have hd : star (Matrix.diagonal (fun i : Fin q => z ^ i.val)) = Matrix.diagonal (fun i : Fin q => star (z ^ i.val)) := by ext i j by_cases h : i = j · subst j; simp [Matrix.star_apply] · simp [Matrix.star_apply, h, Ne.symm h] rw [Matrix.mem_unitaryGroup_iff', hd, Matrix.diagonal_mul_diagonal] simp only [star_pow, ← mul_pow, hz, one_pow] exact Matrix.diagonal_one variable (D : YoungDiagram) (q : ℕ) (hq : D.colLen 0 ≤ q) def weightOrder : ℕ := Fintype.card (Cells D) * q + 1 def weightRoot : ℂ := Complex.exp (2 * Real.pi * Complex.I / weightOrder D q) lemma weightRoot_primitive : IsPrimitiveRoot (weightRoot D q) (weightOrder D q) := Complex.isPrimitiveRoot_exp _ (by unfold weightOrder; omega) lemma weightRoot_unitary : star (weightRoot D q) * weightRoot D q = 1 := by have hn := Complex.norm_eq_one_of_pow_eq_one (weightRoot_primitive D q).pow_eq_one (show weightOrder D q ≠ 0 by unfold weightOrder; omega) rw [mul_comm, Complex.star_def, Complex.mul_conj, Complex.normSq_eq_norm_sq, hn] norm_num def weightElement : Matrix.unitaryGroup (Fin q) ℂ := ⟨Matrix.diagonal (fun i => weightRoot D q ^ i.val), diagonal_unitary (weightRoot_unitary D q)⟩ lemma weight_sum_lt (x : Cells D → Fin q) : ∑ i, (x i).val < weightOrder D q := by have hh : (∑ i, (x i).val) ≤ ∑ _i : Cells D, q := Finset.sum_le_sum fun i _ => (x i).isLt.le simp only [Finset.sum_const, Finset.card_univ, smul_eq_mul] at hh unfold weightOrder omega lemma weight_action (v : TensorSpace D q) (x : Cells D → Fin q) : TensorRep.rep (weightElement D q) v x = weightRoot D q ^ (∑ i, (x i).val) * v x := by change TensorRep.action (Matrix.diagonal (fun i : Fin q => weightRoot D q ^ i.val)) v x = _ rw [TensorRep.action_diagonal, Finset.prod_pow_eq_pow_sum] lemma highest_diagonal (a : Fin q → ℂ) : TensorRep.action (Matrix.diagonal a) (highest D q hq) = (∏ i, a (rowColor D q hq i)) • highest D q hq := by unfold highest rw [PermutationSpace.alternator_apply, map_sum, Finset.smul_sum] apply Finset.sum_congr rfl intro g _ rw [map_smul, action_site_commute, diagonal_single, map_smul] exact smul_comm _ _ _ lemma highest_weight_eigen : TensorRep.rep (weightElement D q) (highest D q hq) = weightRoot D q ^ (∑ i : Cells D, i.val.1) • highest D q hq := by change TensorRep.action (Matrix.diagonal _) _ = _ rw [highest_diagonal] congr 1 exact Finset.prod_pow_eq_pow_sum _ _ _ include hq in lemma weight_eigen_support (v : TensorSpace D q) (hv : TensorRep.rep (weightElement D q) v = weightRoot D q ^ (∑ i : Cells D, i.val.1) • v) (x : Cells D → Fin q) (hx : v x ≠ 0) : ∑ i, (x i).val = ∑ i : Cells D, i.val.1 := by have hh := congrArg (fun w : TensorSpace D q => w x) hv rw [weight_action] at hh simp only [PiLp.smul_apply, smul_eq_mul] at hh have hp := mul_right_cancel₀ hx hh apply (weightRoot_primitive D q).pow_inj (weight_sum_lt D q x) _ hp exact weight_sum_lt D q (rowColor D q hq) theorem weight_eigenspace_one_dimensional (v : TensorSpace D q) (hv : v ∈ alternating D q) (heig : TensorRep.rep (weightElement D q) v = weightRoot D q ^ (∑ i : Cells D, i.val.1) • v) : v = v (rowColor D q hq) • highest D q hq := by ext x change v x = v (rowColor D q hq) * highest D q hq x by_cases hw : ∑ i, (x i).val = ∑ i : Cells D, i.val.1 · exact alternating_minimal_weight D q hq v hv x hw · have hv0 : v x = 0 := by by_contra hn exact hw (weight_eigen_support D q hq v heig x hn) have he0 : highest D q hq x = 0 := by by_contra hn exact hw (weight_eigen_support D q hq _ (highest_weight_eigen D q hq) x hn) rw [hv0, he0, mul_zero] def ordinary : Subrepresentation (alternating D q).toRepresentation := cyclic (alternating D q).toRepresentation (⟨highest D q hq, highest_mem D q hq⟩ : (alternating D q).toSubmodule) theorem ordinary_irreducible : Representation.IsIrreducible (ordinary D q hq).toRepresentation := by refine cyclic_irreducible_of_simple_eigen _ _ ?_ ?_ (weightElement D q) (weightRoot D q ^ (∑ i : Cells D, i.val.1)) ?_ ?_ · intro he exact highest_ne_zero D q hq (congrArg Subtype.val he) · intro g x y exact TensorRep.rep_unitary g (x : TensorSpace D q) (y : TensorSpace D q) · exact Subtype.ext (highest_weight_eigen D q hq) · intro v hv apply Submodule.mem_span_singleton.mpr refine ⟨v.val (rowColor D q hq), ?_⟩ apply Subtype.ext exact (weight_eigenspace_one_dimensional D q hq v.val v.property (congrArg Subtype.val hv)).symm end SpinAngle.PolynomialCarrier namespace SpinAngle.TensorRep open scoped Classical variable {ι κ n : Type*} [contextInstance868_25 : Fintype ι] [contextInstance868_37 : DecidableEq ι] [contextInstance868_53 : Fintype κ] [contextInstance868_65 : DecidableEq κ] [contextInstance869_2 : Fintype n] [contextInstance869_14 : DecidableEq n] lemma diagonal_unitary_of_star_mul (a : n → ℂ) (ha : ∀ i, star (a i) * a i = 1) : Matrix.diagonal a ∈ Matrix.unitaryGroup n ℂ := by have hd : star (Matrix.diagonal a) = Matrix.diagonal (fun i => star (a i)) := by ext i j by_cases h : i = j · subst j; simp [Matrix.star_apply] · simp [Matrix.star_apply, h, Ne.symm h] rw [Matrix.mem_unitaryGroup_iff', hd, Matrix.diagonal_mul_diagonal] simp only [ha] exact Matrix.diagonal_one lemma diagonal_single [DecidableEq (ι → n)] (a : n → ℂ) (b : ι → n) : action (Matrix.diagonal a) (EuclideanSpace.single b (1 : ℂ)) = (∏ i, a (b i)) • EuclideanSpace.single b (1 : ℂ) := by ext x rw [action_diagonal] simp only [PiLp.smul_apply, smul_eq_mul, PiLp.single_apply] by_cases h : x = b · subst x; rfl · simp [h] def content (x : ι → n) (j : n) : ℕ := (Finset.univ.filter fun i => x i = j).card lemma product_content {ι : Type*} {n : Type*} [contextInstance868_25 : Fintype ι] [DecidableEq ι] [contextInstance869_2 : Fintype n] [contextInstance869_14 : DecidableEq n] (a : n → ℂ) (x : ι → n) : ∏ i, a (x i) = ∏ j, a j ^ content x j := by rw [← Finset.prod_fiberwise' Finset.univ x a] simp only [Finset.prod_const, content] lemma phase_product {ι : Type*} {n : Type*} [contextInstance868_25 : Fintype ι] [DecidableEq ι] [Fintype n] [contextInstance869_14 : DecidableEq n] (z : ℂ) (j : n) (x : ι → n) : ∏ i, (if x i = j then z else 1) = z ^ content x j := by rw [← Finset.prod_filter] simp [content] lemma content_le_card {ι : Type*} {n : Type*} [contextInstance868_25 : Fintype ι] [DecidableEq ι] [Fintype n] [contextInstance869_14 : DecidableEq n] (x : ι → n) (j : n) : content x j ≤ Fintype.card ι := (Finset.card_filter_le _ _).trans_eq (Finset.card_univ) lemma intertwiner_content_zero (L : EuclideanSpace ℂ (ι → n) →ₗ[ℂ] EuclideanSpace ℂ (κ → n)) (hL : ∀ (g : Matrix.unitaryGroup n ℂ) v, L (rep g v) = rep g (L v)) (x : ι → n) (y : κ → n) (j : n) (hxy : content x j ≠ content y j) : L (EuclideanSpace.single x (1 : ℂ)) y = 0 := by let N := Fintype.card ι + Fintype.card κ + 1 let z : ℂ := Complex.exp (2 * Real.pi * Complex.I / N) have hz : IsPrimitiveRoot z N := Complex.isPrimitiveRoot_exp _ (by dsimp [N]; omega) have hzn : ‖z‖ = 1 := Complex.norm_eq_one_of_pow_eq_one hz.pow_eq_one (by dsimp [N]; omega) have hzu : star z * z = 1 := by rw [mul_comm, Complex.star_def, Complex.mul_conj, Complex.normSq_eq_norm_sq, hzn] norm_num let g : Matrix.unitaryGroup n ℂ := ⟨Matrix.diagonal (fun k => if k = j then z else 1), diagonal_unitary_of_star_mul _ (fun k => by split_ifs; exact hzu; simp)⟩ have hh := congrArg (fun v : EuclideanSpace ℂ (κ → n) => v y) (hL g (EuclideanSpace.single x (1 : ℂ))) change L (action (Matrix.diagonal _) _) y = action (Matrix.diagonal _) _ y at hh rw [diagonal_single, map_smul, action_diagonal] at hh simp only [PiLp.smul_apply, smul_eq_mul, phase_product] at hh by_contra hn apply hxy exact hz.pow_inj (by have := content_le_card x j; dsimp [N]; omega) (by have := content_le_card y j; dsimp [N]; omega) (mul_right_cancel₀ hn hh) lemma intertwiner_diagonal (L : EuclideanSpace ℂ (ι → n) →ₗ[ℂ] EuclideanSpace ℂ (κ → n)) (hL : ∀ (g : Matrix.unitaryGroup n ℂ) v, L (rep g v) = rep g (L v)) (a : n → ℂ) (v : EuclideanSpace ℂ (ι → n)) : L (action (Matrix.diagonal a) v) = action (Matrix.diagonal a) (L v) := by have hsingle (x : ι → n) : L (action (Matrix.diagonal a) (EuclideanSpace.single x (1 : ℂ))) = action (Matrix.diagonal a) (L (EuclideanSpace.single x (1 : ℂ))) := by ext y rw [diagonal_single, map_smul, action_diagonal] simp only [PiLp.smul_apply, smul_eq_mul] by_cases he : ∀ j, content x j = content y j · rw [product_content a x, product_content a y] congr 1 exact Finset.prod_congr rfl fun j _ => congrArg (fun k => a j ^ k) (he j) · push Not at he obtain ⟨j, hj⟩ := he rw [intertwiner_content_zero L hL x y j hj, mul_zero, mul_zero] have hexp := (EuclideanSpace.basisFun (ι → n) ℂ).sum_repr v simp only [EuclideanSpace.basisFun_repr, EuclideanSpace.basisFun_apply] at hexp rw [← hexp, map_sum, map_sum, map_sum, map_sum] apply Finset.sum_congr rfl intro x _ rw [map_smul, map_smul, map_smul, map_smul, hsingle] lemma intertwiner_hermitian (L : EuclideanSpace ℂ (ι → n) →ₗ[ℂ] EuclideanSpace ℂ (κ → n)) (hL : ∀ (g : Matrix.unitaryGroup n ℂ) v, L (rep g v) = rep g (L v)) (a : Matrix n n ℂ) (ha : a.IsHermitian) (v : EuclideanSpace ℂ (ι → n)) : L (action a v) = action a (L v) := by rw [ha.spectral_theorem] simp only [Unitary.conjStarAlgAut_apply, map_mul, Module.End.mul_apply] change L (rep ha.eigenvectorUnitary (action (Matrix.diagonal _) (rep (star ha.eigenvectorUnitary) v))) = _ rw [hL, intertwiner_diagonal L hL, hL] rfl end SpinAngle.TensorRep namespace SpinAngle.TensorRep open SpinAngle.Specht variable {ι n : Type*} [contextInstance980_23 : Fintype ι] [contextInstance980_35 : DecidableEq ι] [contextInstance980_51 : Fintype n] [contextInstance980_63 : DecidableEq n] lemma apply_mem_hermitian (U : Subrepresentation (rep (ι := ι) (n := n))) (a : Matrix n n ℂ) (ha : a.IsHermitian) (v : EuclideanSpace ℂ (ι → n)) (hv : v ∈ U) : action a v ∈ U := by let L : EuclideanSpace ℂ (ι → n) →ₗ[ℂ] EuclideanSpace ℂ (ι → n) := U.toSubmodule.starProjection.toLinearMap have hu : ∀ (g : Matrix.unitaryGroup n ℂ) (x y : EuclideanSpace ℂ (ι → n)), ⟪rep g x, rep g y⟫_ℂ = ⟪x, y⟫_ℂ := rep_unitary have hL : ∀ (g : Matrix.unitaryGroup n ℂ) (w : EuclideanSpace ℂ (ι → n)), L (rep g w) = rep g (L w) := by intro g w exact starProjection_rep (rep (ι := ι) (n := n)) hu U g w have hlv : L v = v := U.toSubmodule.starProjection_eq_self_iff.mpr hv have hh := intertwiner_hermitian L hL a ha v rw [hlv] at hh rw [← hh] exact U.toSubmodule.starProjection_apply_mem _ end SpinAngle.TensorRep namespace SpinAngle.PolynomialCarrier open scoped Classical Matrix open SpinAngle.Specht SpinAngle.Young variable (D : YoungDiagram) (q : ℕ) (hq : D.colLen 0 ≤ q) def highestMonomial (t : Fin q → ℝ) : ℝ := ∏ i : Cells D, t (rowColor D q hq i) lemma highestMonomial_nonneg (t : Fin q → ℝ) (ht : ∀ j, 0 ≤ t j) : 0 ≤ highestMonomial D q hq t := Finset.prod_nonneg fun _ _ => ht _ lemma monomial_le_highest (t : Fin q → ℝ) (ht : ∀ j, 0 ≤ t j) (hanti : Antitone t) (x : Cells D → Fin q) (hinj : ∀ i j : Cells D, i.val.2 = j.val.2 → (x i).val = (x j).val → i = j) : (∏ i, t (x i)) ≤ highestMonomial D q hq t := by obtain ⟨σ, -, hσ⟩ := column_sort_le D (fun i => (x i).val) hinj rw [← Equiv.prod_comp σ (fun i => t (x i))] apply Finset.prod_le_prod₀ (fun _ _ => ht _) intro i _ exact hanti (show rowColor D q hq i ≤ x (σ i) from hσ i) theorem diagonal_norm_bound (t : Fin q → ℝ) (ht : ∀ j, 0 ≤ t j) (hanti : Antitone t) (v : TensorSpace D q) (hv : v ∈ alternating D q) : ‖TensorRep.action (Matrix.diagonal (fun j => (t j : ℂ))) v‖ ≤ highestMonomial D q hq t * ‖v‖ := by have hb := highestMonomial_nonneg D q hq t ht apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hb (norm_nonneg _))).mp rw [mul_pow, EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq, Finset.mul_sum] apply Finset.sum_le_sum intro x _ rw [TensorRep.action_diagonal, norm_mul, norm_prod, mul_pow] simp only [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (ht _)] by_cases hz : v x = 0 · simp [hz] · apply mul_le_mul_of_nonneg_right _ (sq_nonneg _) exact pow_le_pow_left₀ (Finset.prod_nonneg fun _ _ => ht _) (monomial_le_highest D q hq t ht hanti x (alternating_injective D q v hv x hz)) 2 end SpinAngle.PolynomialCarrier noncomputable section open scoped Matrix ComplexOrder namespace SpinAngle.SortedSpectrum variable {q : ℕ} (A : Matrix (Fin q) (Fin q) ℂ) (hA : A.IsHermitian) def values : Fin q → ℝ := (Matrix.isSymmetric_toEuclideanLin_iff.mpr hA).eigenvalues (by simp) def basis : OrthonormalBasis (Fin q) ℂ (EuclideanSpace ℂ (Fin q)) := (Matrix.isSymmetric_toEuclideanLin_iff.mpr hA).eigenvectorBasis (by simp) def changeBasis : Matrix.unitaryGroup (Fin q) ℂ := ⟨(EuclideanSpace.basisFun (Fin q) ℂ).toBasis.toMatrix (basis A hA).toBasis, (EuclideanSpace.basisFun (Fin q) ℂ).toMatrix_orthonormalBasis_mem_unitary (basis A hA)⟩ @[simp] lemma changeBasis_apply (i j : Fin q) : changeBasis A hA i j = basis A hA j i := rfl lemma values_antitone : Antitone (values A hA) := LinearMap.IsSymmetric.eigenvalues_antitone _ _ lemma eigenvector (i : Fin q) : Matrix.toEuclideanLin A (basis A hA i) = (values A hA i : ℂ) • basis A hA i := LinearMap.IsSymmetric.apply_eigenvectorBasis _ _ _ lemma spectral_theorem : A = (changeBasis A hA : Matrix _ _ ℂ) * Matrix.diagonal (fun i => (values A hA i : ℂ)) * star (changeBasis A hA : Matrix _ _ ℂ) := by have he : A * (changeBasis A hA : Matrix _ _ ℂ) = (changeBasis A hA : Matrix _ _ ℂ) * Matrix.diagonal (fun i => (values A hA i : ℂ)) := by ext i j rw [Matrix.mul_diagonal] simpa only [Matrix.mul_apply, changeBasis_apply, Matrix.toLpLin_apply, Matrix.mulVec, dotProduct, PiLp.smul_apply, smul_eq_mul, mul_comm] using congrArg (fun v : EuclideanSpace ℂ (Fin q) => v i) (eigenvector A hA j) have hh := congrArg (fun B : Matrix (Fin q) (Fin q) ℂ => B * star (changeBasis A hA : Matrix _ _ ℂ)) he simpa only [Matrix.mul_assoc, (changeBasis A hA).property.2, Matrix.mul_one] using hh lemma values_nonneg (hpos : A.PosSemidef) (i : Fin q) : 0 ≤ values A hA i := by have hp := (Matrix.isPositive_toEuclideanLin_iff.mpr hpos).inner_nonneg_left (basis A hA i) rw [eigenvector, inner_smul_left, (basis A hA).inner_eq_one i] at hp have hp' : (0 : ℂ) ≤ (values A hA i : ℂ) := by simpa only [Complex.conj_ofReal, mul_one] using hp exact RCLike.ofReal_nonneg.mp hp' lemma trace_eq_sum : Matrix.trace A = ∑ i, (values A hA i : ℂ) := by conv_lhs => rw [spectral_theorem A hA] rw [Matrix.trace_mul_cycle, (changeBasis A hA).property.1, Matrix.one_mul, Matrix.trace_diagonal] end SpinAngle.SortedSpectrum namespace SpinAngle.PolynomialCarrier open scoped Classical Matrix InnerProductSpace open SpinAngle.Specht SpinAngle.Young Module variable (D : YoungDiagram) (q : ℕ) (hq : D.colLen 0 ≤ q) abbrev Carrier : Type := (ordinary D q hq).toSubmodule instance carrierInner : InnerProductSpace ℂ (Carrier D q hq) := @Submodule.innerProductSpace ℂ (alternating D q).toSubmodule _ _ inferInstance (ordinary D q hq).toSubmodule def inclusion : Carrier D q hq →ₗᵢ[ℂ] TensorSpace D q := (alternating D q).toSubmodule.subtypeₗᵢ.comp (ordinary D q hq).toSubmodule.subtypeₗᵢ @[simp] lemma inclusion_apply (v : Carrier D q hq) : inclusion D q hq v = v.val.val := rfl lemma inclusion_equivariant (g : Matrix.unitaryGroup (Fin q) ℂ) (v : Carrier D q hq) : inclusion D q hq ((ordinary D q hq).toRepresentation g v) = TensorRep.rep g (inclusion D q hq v) := rfl def image : Subrepresentation (TensorRep.rep (ι := Cells D) (n := Fin q)) where toSubmodule := LinearMap.range (inclusion D q hq).toLinearMap apply_mem_toSubmodule := by rintro g v ⟨x,rfl⟩ exact ⟨(ordinary D q hq).toRepresentation g x,rfl⟩ lemma inclusion_image (v : Carrier D q hq) : inclusion D q hq v ∈ image D q hq := ⟨v,rfl⟩ def hermitianAction (a : Matrix (Fin q) (Fin q) ℂ) (ha : a.IsHermitian) : Carrier D q hq →ₗ[ℂ] Carrier D q hq := (LinearEquiv.ofInjective (inclusion D q hq).toLinearMap (inclusion D q hq).injective).symm.toLinearMap.comp ((TensorRep.action a).comp (inclusion D q hq).toLinearMap |>.codRestrict (image D q hq).toSubmodule (fun v => TensorRep.apply_mem_hermitian _ a ha _ (inclusion_image D q hq v))) @[simp] lemma inclusion_hermitianAction (a : Matrix (Fin q) (Fin q) ℂ) (ha : a.IsHermitian) (v : Carrier D q hq) : inclusion D q hq (hermitianAction D q hq a ha v) = TensorRep.action a (inclusion D q hq v) := by exact congrArg Subtype.val ((LinearEquiv.ofInjective (inclusion D q hq).toLinearMap (inclusion D q hq).injective).apply_symm_apply _) lemma tensor_hermitian (a : Matrix (Fin q) (Fin q) ℂ) (ha : a.IsHermitian) : (TensorRep.tensor (fun _ : Cells D => a)).IsHermitian := by change star (TensorRep.tensor (fun _ : Cells D => a)) = _ rw [← TensorRep.tensor_star] congr 1 funext i exact ha lemma hermitianAction_symmetric (a : Matrix (Fin q) (Fin q) ℂ) (ha : a.IsHermitian) : LinearMap.IsSymmetric (𝕜 := ℂ) (E := Carrier D q hq) (hermitianAction D q hq a ha) := by intro x y rw [← (inclusion D q hq).inner_map_map, ← (inclusion D q hq).inner_map_map x, inclusion_hermitianAction, inclusion_hermitianAction] exact (Matrix.isSymmetric_toEuclideanLin_iff.mpr (tensor_hermitian D q a ha)) x.val.val y.val.val lemma ordinary_rep_unitary (g : Matrix.unitaryGroup (Fin q) ℂ) (v w : Carrier D q hq) : ⟪(ordinary D q hq).toRepresentation g v, (ordinary D q hq).toRepresentation g w⟫_ℂ = ⟪v,w⟫_ℂ := TensorRep.rep_unitary g v.val.val w.val.val lemma raw_norm_unitary (g : Matrix.unitaryGroup (Fin q) ℂ) (v : TensorSpace D q) : ‖TensorRep.rep g v‖ = ‖v‖ := (TensorRep.rep g).isometryOfInner (TensorRep.rep_unitary g) |>.norm_map v theorem hermitianAction_norm_bound (a : Matrix (Fin q) (Fin q) ℂ) (ha : a.PosSemidef) (v : Carrier D q hq) : ‖hermitianAction D q hq a ha.isHermitian v‖ ≤ highestMonomial D q hq (SortedSpectrum.values a ha.isHermitian) * ‖v‖ := by rw [← (inclusion D q hq).norm_map (hermitianAction D q hq a ha.isHermitian v), inclusion_hermitianAction, ← (inclusion D q hq).norm_map v] let u := SortedSpectrum.changeBasis a ha.isHermitian let t := SortedSpectrum.values a ha.isHermitian have he := SortedSpectrum.spectral_theorem a ha.isHermitian have hin : TensorRep.rep u⁻¹ (inclusion D q hq v) ∈ alternating D q := (alternating D q).apply_mem_toSubmodule u⁻¹ v.val.property have hb := diagonal_norm_bound D q hq t (SortedSpectrum.values_nonneg a ha.isHermitian ha) (SortedSpectrum.values_antitone a ha.isHermitian) _ hin change a = (u : Matrix _ _ ℂ) * Matrix.diagonal (fun i => (t i : ℂ)) * (↑(u⁻¹) : Matrix _ _ ℂ) at he conv_lhs => rw [he] simp only [map_mul, Module.End.mul_apply] change ‖TensorRep.rep u (TensorRep.action (Matrix.diagonal _) (TensorRep.rep u⁻¹ _))‖ ≤ _ rw [raw_norm_unitary] simpa only [raw_norm_unitary] using hb end SpinAngle.PolynomialCarrier noncomputable section open scoped TensorProduct InnerProductSpace BigOperators namespace SpinAngle.ExternalTensor open Module variable {E F : Type*} [contextInstance1195_23 : NormedAddCommGroup E] [contextInstance1195_46 : InnerProductSpace ℂ E] [contextInstance1196_2 : NormedAddCommGroup F] [contextInstance1196_25 : InnerProductSpace ℂ F] [contextInstance1197_2 : FiniteDimensional ℂ E] [contextInstance1197_26 : FiniteDimensional ℂ F] variable {G H : Type*} [contextInstance1198_23 : Group G] [contextInstance1198_33 : Group H] def rep (ρ : Representation ℂ G E) (σ : Representation ℂ H F) : Representation ℂ (G × H) (E ⊗[ℂ] F) := Representation.tprod (ρ.comp (MonoidHom.fst G H)) (σ.comp (MonoidHom.snd G H)) @[simp] lemma rep_tmul {E : Type*} {F : Type*} [contextInstance1195_23 : NormedAddCommGroup E] [contextInstance1195_46 : InnerProductSpace ℂ E] [contextInstance1196_2 : NormedAddCommGroup F] [contextInstance1196_25 : InnerProductSpace ℂ F] [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] {G : Type*} {H : Type*} [contextInstance1198_23 : Group G] [contextInstance1198_33 : Group H] (ρ : Representation ℂ G E) (σ : Representation ℂ H F) (g : G × H) (x : E) (y : F) : rep ρ σ g (x ⊗ₜ[ℂ] y) = ρ g.1 x ⊗ₜ σ g.2 y := rfl def rightSlice (β : Module.Dual ℂ F) : E ⊗[ℂ] F →ₗ[ℂ] E := (TensorProduct.rid ℂ E).toLinearMap.comp (TensorProduct.map LinearMap.id β) def leftSlice (α : Module.Dual ℂ E) : E ⊗[ℂ] F →ₗ[ℂ] F := (TensorProduct.lid ℂ F).toLinearMap.comp (TensorProduct.map α LinearMap.id) @[simp] lemma rightSlice_tmul {E : Type*} {F : Type*} [contextInstance1195_23 : NormedAddCommGroup E] [contextInstance1195_46 : InnerProductSpace ℂ E] [contextInstance1196_2 : NormedAddCommGroup F] [contextInstance1196_25 : InnerProductSpace ℂ F] [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] (β : Module.Dual ℂ F) (x : E) (y : F) : rightSlice β (x ⊗ₜ[ℂ] y) = β y • x := rfl @[simp] lemma leftSlice_tmul {E : Type*} {F : Type*} [contextInstance1195_23 : NormedAddCommGroup E] [contextInstance1195_46 : InnerProductSpace ℂ E] [contextInstance1196_2 : NormedAddCommGroup F] [contextInstance1196_25 : InnerProductSpace ℂ F] [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] (α : Module.Dual ℂ E) (x : E) (y : F) : leftSlice α (x ⊗ₜ[ℂ] y) = α x • y := rfl lemma slice_commute {E : Type*} {F : Type*} [contextInstance1195_23 : NormedAddCommGroup E] [contextInstance1195_46 : InnerProductSpace ℂ E] [contextInstance1196_2 : NormedAddCommGroup F] [contextInstance1196_25 : InnerProductSpace ℂ F] [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] (α : Module.Dual ℂ E) (β : Module.Dual ℂ F) (v : E ⊗[ℂ] F) : α (rightSlice β v) = β (leftSlice α v) := by induction v using TensorProduct.inductionOn with | tmul x y => simp [mul_comm] | add x y hx hy => simp [hx, hy] lemma slice_map_left {E : Type*} {F : Type*} [contextInstance1195_23 : NormedAddCommGroup E] [contextInstance1195_46 : InnerProductSpace ℂ E] [contextInstance1196_2 : NormedAddCommGroup F] [contextInstance1196_25 : InnerProductSpace ℂ F] [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] (β : Module.Dual ℂ F) (A : E →ₗ[ℂ] E) (v : E ⊗[ℂ] F) : rightSlice β (TensorProduct.map A LinearMap.id v) = A (rightSlice β v) := by induction v using TensorProduct.inductionOn with | tmul x y => simp | add x y hx hy => simp [hx, hy] lemma eq_sum_slices {E : Type*} {F : Type*} [contextInstance1195_23 : NormedAddCommGroup E] [contextInstance1195_46 : InnerProductSpace ℂ E] [contextInstance1196_2 : NormedAddCommGroup F] [contextInstance1196_25 : InnerProductSpace ℂ F] [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] {ι : Type*} [Fintype ι] (b : Basis ι ℂ F) (v : E ⊗[ℂ] F) : v = ∑ i, rightSlice (b.coord i) v ⊗ₜ[ℂ] b i := by induction v using TensorProduct.inductionOn with | tmul x y => simp only [rightSlice_tmul, TensorProduct.smul_tmul, ← TensorProduct.tmul_sum, Basis.coord_apply, Basis.sum_repr] | add x y hx hy => simp only [map_add, TensorProduct.add_tmul, Finset.sum_add_distrib, ← hx, ← hy] lemma rightSlice_ext {v w : E ⊗[ℂ] F} (h : ∀ β : Module.Dual ℂ F, rightSlice β v = rightSlice β w) : v = w := by classical let b := Module.finBasis ℂ F rw [eq_sum_slices b v, eq_sum_slices b w] exact Finset.sum_congr rfl fun i _ => congrArg (fun x : E => x ⊗ₜ[ℂ] b i) (h _) lemma scalar_of_commute (ρ : Representation ℂ G E) [ρ.IsIrreducible] (A : E →ₗ[ℂ] E) (hA : ∀ g x, A (ρ g x) = ρ g (A x)) : ∃ z : ℂ, A = z • LinearMap.id := by let f := A.intertwiningMap_of_isIntertwiningMap ρ ρ hA obtain ⟨z, hz⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed (ρ := ρ)).surjective f refine ⟨z, ?_⟩ ext x exact (congrArg (fun L : ρ.IntertwiningMap ρ => L x) hz).symm lemma left_commutant (ρ : Representation ℂ G E) [ρ.IsIrreducible] (d : E ⊗[ℂ] F →ₗ[ℂ] E ⊗[ℂ] F) (hd : ∀ g v, d (TensorProduct.map (ρ g) LinearMap.id v) = TensorProduct.map (ρ g) LinearMap.id (d v)) : ∃ B : F →ₗ[ℂ] F, d = TensorProduct.map LinearMap.id B := by classical let : Nontrivial E := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) ρ.asModule obtain ⟨x₀, hx₀⟩ := exists_ne (0 : E) obtain ⟨α, hα⟩ := Module.Projective.exists_dual_eq_one ℂ hx₀ let B : F →ₗ[ℂ] F := (leftSlice α).comp (d.comp (TensorProduct.mk ℂ E F x₀)) refine ⟨B, ?_⟩ ext x y change d (x ⊗ₜ[ℂ] y) = TensorProduct.map LinearMap.id B (x ⊗ₜ[ℂ] y) apply rightSlice_ext intro β let A : E →ₗ[ℂ] E := (rightSlice β).comp (d.comp ((TensorProduct.mk ℂ E F).flip y)) have hA (g : G) (v : E) : A (ρ g v) = ρ g (A v) := by change rightSlice β (d (ρ g v ⊗ₜ[ℂ] y)) = ρ g (rightSlice β (d (v ⊗ₜ[ℂ] y))) rw [← slice_map_left, ← hd] rfl obtain ⟨z,hz⟩ := scalar_of_commute ρ A hA have hzv (v : E) : rightSlice β (d (v ⊗ₜ[ℂ] y)) = z • v := congrArg (fun f : E →ₗ[ℂ] E => f v) hz have hzB : β (B y) = z := by change β (leftSlice α (d (x₀ ⊗ₜ[ℂ] y))) = z rw [← slice_commute, hzv, map_smul, hα, smul_eq_mul, mul_one] change rightSlice β (d (x ⊗ₜ[ℂ] y)) = β (B y) • x rw [hzv, hzB] lemma commutant_scalar (ρ : Representation ℂ G E) (σ : Representation ℂ H F) [ρ.IsIrreducible] [σ.IsIrreducible] (d : (rep ρ σ).IntertwiningMap (rep ρ σ)) : ∃ z : ℂ, d.toLinearMap = z • LinearMap.id := by obtain ⟨B,hB⟩ := left_commutant ρ d.toLinearMap (by intro g v simpa [rep, Representation.tprod_apply, Module.End.one_eq_id] using Representation.IntertwiningMap.isIntertwining _ _ d (g,1) v) have hσ (h : H) (y : F) : B (σ h y) = σ h (B y) := by let : Nontrivial E := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) ρ.asModule obtain ⟨x,hx⟩ := exists_ne (0 : E) obtain ⟨α,hα⟩ := Module.Projective.exists_dual_eq_one ℂ hx have hh := Representation.IntertwiningMap.isIntertwining _ _ d (1,h) (x ⊗ₜ[ℂ] y) change d.toLinearMap (rep ρ σ (1,h) (x ⊗ₜ y)) = rep ρ σ (1,h) (d.toLinearMap (x ⊗ₜ y)) at hh rw [hB] at hh simp only [rep_tmul, map_one, Module.End.one_apply, TensorProduct.map_tmul, LinearMap.id_apply] at hh have := congrArg (leftSlice α) hh simpa [hα] using this obtain ⟨z,hz⟩ := scalar_of_commute σ B hσ refine ⟨z, ?_⟩ rw [hB, hz] ext x y simp lemma unitary {E : Type*} {F : Type*} [contextInstance1195_23 : NormedAddCommGroup E] [contextInstance1195_46 : InnerProductSpace ℂ E] [contextInstance1196_2 : NormedAddCommGroup F] [contextInstance1196_25 : InnerProductSpace ℂ F] [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] {G : Type*} {H : Type*} [contextInstance1198_23 : Group G] [contextInstance1198_33 : Group H] (ρ : Representation ℂ G E) (σ : Representation ℂ H F) (hρ : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x,y⟫_ℂ) (hσ : ∀ h x y, ⟪σ h x, σ h y⟫_ℂ = ⟪x,y⟫_ℂ) (g : G × H) (v w : E ⊗[ℂ] F) : ⟪rep ρ σ g v, rep ρ σ g w⟫_ℂ = ⟪v,w⟫_ℂ := by let a : E →ₗᵢ[ℂ] E := (ρ g.1).isometryOfInner (hρ g.1) let b : F →ₗᵢ[ℂ] F := (σ g.2).isometryOfInner (hσ g.2) exact TensorProduct.inner_map_map a b v w lemma projection_commute (τ : Representation ℂ G E) (hτ : ∀ g x y, ⟪τ g x, τ g y⟫_ℂ = ⟪x,y⟫_ℂ) (U : Subrepresentation τ) (g : G) (v : E) : U.toSubmodule.starProjection (τ g v) = τ g (U.toSubmodule.starProjection v) := by apply U.toSubmodule.eq_starProjection_of_mem_of_inner_eq_zero · exact U.apply_mem_toSubmodule g (U.toSubmodule.starProjection_apply_mem v) · intro w hw have hw' := U.apply_mem_toSubmodule g⁻¹ hw have he : τ g (τ g⁻¹ w) = w := by rw [← Module.End.mul_apply, ← map_mul, mul_inv_cancel, map_one, Module.End.one_apply] rw [← map_sub] conv_lhs => rhs; rw [← he] rw [hτ] exact U.toSubmodule.starProjection_inner_eq_zero v _ hw' theorem irreducible (ρ : Representation ℂ G E) (σ : Representation ℂ H F) [ρ.IsIrreducible] [σ.IsIrreducible] (hρ : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x,y⟫_ℂ) (hσ : ∀ h x y, ⟪σ h x, σ h y⟫_ℂ = ⟪x,y⟫_ℂ) : (rep ρ σ).IsIrreducible := by let : Nontrivial E := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) ρ.asModule let : Nontrivial F := IsSimpleModule.nontrivial (MonoidAlgebra ℂ H) σ.asModule obtain ⟨x,hx⟩ := exists_ne (0 : E) obtain ⟨y,hy⟩ := exists_ne (0 : F) have hxy : x ⊗ₜ[ℂ] y ≠ 0 := by intro h have := congrArg norm h simp only [TensorProduct.norm_tmul, norm_zero] at this exact (mul_ne_zero (norm_ne_zero_iff.mpr hx) (norm_ne_zero_iff.mpr hy)) this have htb : (⊤ : Subrepresentation (rep ρ σ)) ≠ ⊥ := by intro h have hh : x ⊗ₜ[ℂ] y ∈ (⊤ : Subrepresentation (rep ρ σ)) := by trivial rw [h] at hh exact hxy hh let : Nontrivial (Subrepresentation (rep ρ σ)) := ⟨⟨⊤,⊥,htb⟩⟩ refine ⟨fun U => ?_⟩ let d : (rep ρ σ).IntertwiningMap (rep ρ σ) := U.toSubmodule.starProjection.toLinearMap.intertwiningMap_of_isIntertwiningMap _ _ (projection_commute _ (unitary ρ σ hρ hσ) U) obtain ⟨z,hz⟩ := commutant_scalar ρ σ d have hproj (v : E ⊗[ℂ] F) : U.toSubmodule.starProjection v = z • v := congrArg (fun f : E ⊗[ℂ] F →ₗ[ℂ] E ⊗[ℂ] F => f v) hz by_cases hz0 : z = 0 · left apply bot_unique intro v hv change v = 0 have hh := hproj v rw [U.toSubmodule.starProjection_eq_self_iff.mpr hv, hz0, zero_smul] at hh exact hh · right apply top_unique intro v _ have hv := U.toSubmodule.starProjection_apply_mem (z⁻¹ • v) rw [hproj, smul_smul, mul_inv_cancel₀ hz0, one_smul] at hv exact hv end SpinAngle.ExternalTensor noncomputable section open scoped BigOperators ComplexConjugate namespace SpinAngle.TorusNaturality open Module variable {n X Y E F : Type*} [contextInstance1400_29 : Fintype n] [contextInstance1400_41 : Fintype X] [contextInstance1400_53 : Fintype Y] [contextInstance1401_2 : AddCommGroup E] [contextInstance1401_19 : Module ℂ E] [contextInstance1401_32 : AddCommGroup F] [contextInstance1401_49 : Module ℂ F] variable (b : Basis X ℂ E) (c : Basis Y ℂ F) (w : X → n → ℕ) (v : Y → n → ℕ) (D : (n → ℂ) → E →ₗ[ℂ] E) (D' : (n → ℂ) → F →ₗ[ℂ] F) (hb : ∀ a x, D a (b x) = (∏ j, a j ^ w x j) • b x) (hc : ∀ a y, D' a (c y) = (∏ j, a j ^ v y j) • c y) include hc in lemma repr_diagonal (a : n → ℂ) (f : F) (y : Y) : c.repr (D' a f) y = (∏ j, a j ^ v y j) * c.repr f y := by classical calc c.repr (D' a f) y = c.repr (D' a (∑ k, c.repr f k • c k)) y := by rw [c.sum_repr] _ = _ := by simp only [map_sum, map_smul, hc, Basis.repr_self] simp [Finset.sum_apply, Finsupp.single_apply, mul_comm] lemma coefficient_zero {n : Type*} {X : Type*} {Y : Type*} {E : Type*} {F : Type*} [contextInstance1400_29 : Fintype n] [Fintype X] [contextInstance1400_53 : Fintype Y] [contextInstance1401_2 : AddCommGroup E] [contextInstance1401_19 : Module ℂ E] [contextInstance1401_32 : AddCommGroup F] [contextInstance1401_49 : Module ℂ F] (b : Basis X ℂ E) (c : Basis Y ℂ F) (w : X → n → ℕ) (v : Y → n → ℕ) (D : (n → ℂ) → E →ₗ[ℂ] E) (D' : (n → ℂ) → F →ₗ[ℂ] F) (hb : ∀ a x, D a (b x) = (∏ j, a j ^ w x j) • b x) (hc : ∀ a y, D' a (c y) = (∏ j, a j ^ v y j) • c y) (L : E →ₗ[ℂ] F) (hL : ∀ a : n → ℂ, (∀ j, star (a j) * a j = 1) → ∀ x, L (D a x) = D' a (L x)) (x : X) (y : Y) (j : n) (hxy : w x j ≠ v y j) : c.repr (L (b x)) y = 0 := by classical let N := w x j + v y j + 1 let z : ℂ := Complex.exp (2 * Real.pi * Complex.I / N) have hN : N ≠ 0 := by dsimp [N]; omega have hz : IsPrimitiveRoot z N := Complex.isPrimitiveRoot_exp _ hN have hzn : ‖z‖ = 1 := Complex.norm_eq_one_of_pow_eq_one hz.pow_eq_one hN have hzu : star z * z = 1 := by rw [mul_comm, Complex.star_def, Complex.mul_conj, Complex.normSq_eq_norm_sq, hzn] norm_num let a : n → ℂ := fun k => if k = j then z else 1 have ha : ∀ k, star (a k) * a k = 1 := by intro k dsimp [a] split_ifs · exact hzu · simp have hh := congrArg (fun f => c.repr f y) (hL a ha (b x)) rw [hb, map_smul, map_smul, Finsupp.smul_apply, smul_eq_mul, repr_diagonal c v D' hc] at hh have hprod (u : n → ℕ) : (∏ k, a k ^ u k) = z ^ u j := by simp [a] rw [hprod, hprod] at hh by_contra hn exact hxy (hz.pow_inj (by dsimp [N]; omega) (by dsimp [N]; omega) (mul_right_cancel₀ hn hh)) include b c w v hb hc in theorem diagonal (L : E →ₗ[ℂ] F) (hL : ∀ a : n → ℂ, (∀ j, star (a j) * a j = 1) → ∀ x, L (D a x) = D' a (L x)) (a : n → ℂ) : L.comp (D a) = (D' a).comp L := by classical apply b.ext intro x apply c.repr.injective ext y simp only [LinearMap.comp_apply, hb, map_smul, Finsupp.smul_apply, smul_eq_mul, repr_diagonal c v D' hc] by_cases hw : w x = v y · rw [hw] · obtain ⟨j,hj⟩ := Function.ne_iff.mp hw rw [coefficient_zero b c w v D D' hb hc L hL x y j hj] simp end SpinAngle.TorusNaturality namespace SpinAngle.SignedTensor open scoped Classical Matrix TensorProduct InnerProductSpace open Module variable {q : ℕ} abbrev Spin (q : ℕ) := Fin q × Bool abbrev Mat (q : ℕ) := Matrix (Fin q) (Fin q) ℂ abbrev Group (q : ℕ) := Matrix.unitaryGroup (Fin q) ℂ × Matrix.unitaryGroup (Fin q) ℂ def block : (Mat q × Mat q) →* Matrix (Spin q) (Spin q) ℂ where toFun a := Matrix.blockDiagonal (fun b : Bool => cond b a.2 a.1) map_one' := by convert (Matrix.blockDiagonal_one (m := Fin q) (o := Bool) (α := ℂ)) using 1 congr 1 funext b cases b <;> rfl map_mul' a b := by rw [← Matrix.blockDiagonal_mul] congr 1 funext c cases c <;> rfl lemma block_star (a : Mat q × Mat q) : block (star a) = star (block a) := by change Matrix.blockDiagonal (fun b : Bool => cond b (star a.2) (star a.1)) = (Matrix.blockDiagonal (fun b : Bool => cond b a.2 a.1))ᴴ rw [Matrix.blockDiagonal_conjTranspose] congr 1 funext b cases b <;> rfl def blockGroup : Group q →* Matrix.unitaryGroup (Spin q) ℂ where toFun g := ⟨block (g.1.val,g.2.val), by rw [Matrix.mem_unitaryGroup_iff', ← block_star, ← map_mul] have he : star (g.1.val,g.2.val) * (g.1.val,g.2.val) = 1 := Prod.ext g.1.property.1 g.2.property.1 rw [he, map_one]⟩ map_one' := by apply Subtype.ext change block (1 : Mat q × Mat q) = 1 exact block.map_one map_mul' g h := by apply Subtype.ext change block ((g.1.val,g.2.val) * (h.1.val,h.2.val)) = block (g.1.val,g.2.val) * block (h.1.val,h.2.val) exact block.map_mul _ _ variable {ι κ τ : Type*} [contextInstance1517_25 : Fintype ι] [contextInstance1517_37 : DecidableEq ι] [contextInstance1517_53 : Fintype κ] [contextInstance1517_65 : DecidableEq κ] [contextInstance1518_2 : Fintype τ] [contextInstance1518_14 : DecidableEq τ] abbrev Raw := EuclideanSpace ℂ (τ → Spin q) abbrev PairRaw := EuclideanSpace ℂ (ι → Fin q) ⊗[ℂ] EuclideanSpace ℂ (κ → Fin q) def rep : Representation ℂ (Group q) (Raw (q := q) (τ := τ)) := (TensorRep.rep (ι := τ)).comp blockGroup def pairRep : Representation ℂ (Group q) (PairRaw (q := q) (ι := ι) (κ := κ)) := ExternalTensor.rep TensorRep.rep TensorRep.rep def pairAction : (Mat q × Mat q) →* Module.End ℂ (PairRaw (q := q) (ι := ι) (κ := κ)) where toFun a := TensorProduct.map (TensorRep.action a.1) (TensorRep.action a.2) map_one' := by ext x y change TensorProduct.map _ _ (x ⊗ₜ[ℂ] y) = x ⊗ₜ[ℂ] y simp only [Prod.fst_one, Prod.snd_one, map_one, TensorProduct.map_tmul, Module.End.one_apply] map_mul' a b := by ext x y change TensorProduct.map _ _ (x ⊗ₜ[ℂ] y) = TensorProduct.map _ _ (TensorProduct.map _ _ (x ⊗ₜ[ℂ] y)) simp only [Prod.fst_mul, Prod.snd_mul, map_mul, TensorProduct.map_tmul, Module.End.mul_apply] @[simp] lemma pairAction_tmul (a : Mat q × Mat q) (x : EuclideanSpace ℂ (ι → Fin q)) (y : EuclideanSpace ℂ (κ → Fin q)) : pairAction a (x ⊗ₜ[ℂ] y) = TensorRep.action a.1 x ⊗ₜ TensorRep.action a.2 y := rfl lemma pairRep_eq (g : Group q) : pairRep (ι := ι) (κ := κ) g = pairAction (g.1.val,g.2.val) := rfl lemma rep_unitary (g : Group q) (v w : Raw (q := q) (τ := τ)) : ⟪rep g v,rep g w⟫_ℂ = ⟪v,w⟫_ℂ := TensorRep.rep_unitary (blockGroup g) v w lemma pairRep_unitary (g : Group q) (v w : PairRaw (q := q) (ι := ι) (κ := κ)) : ⟪pairRep g v,pairRep g w⟫_ℂ = ⟪v,w⟫_ℂ := ExternalTensor.unitary (TensorRep.rep (ι := ι) (n := Fin q)) (TensorRep.rep (ι := κ) (n := Fin q)) TensorRep.rep_unitary TensorRep.rep_unitary g v w def pairBasis : OrthonormalBasis ((ι → Fin q) × (κ → Fin q)) ℂ (PairRaw (q := q) (ι := ι) (κ := κ)) := (EuclideanSpace.basisFun _ ℂ).tensorProduct (EuclideanSpace.basisFun _ ℂ) def pairContent (x : (ι → Fin q) × (κ → Fin q)) (j : Spin q) : ℕ := cond j.2 (TensorRep.content x.2 j.1) (TensorRep.content x.1 j.1) def diagPair (a : Spin q → ℂ) : Mat q × Mat q := (Matrix.diagonal (fun j => a (j,false)), Matrix.diagonal (fun j => a (j,true))) lemma block_diagPair (a : Spin q → ℂ) : block (diagPair a) = Matrix.diagonal a := by change Matrix.blockDiagonal (fun b : Bool => cond b (Matrix.diagonal (fun j => a (j,true))) (Matrix.diagonal (fun j => a (j,false)))) = Matrix.diagonal a convert Matrix.blockDiagonal_diagonal (fun (b : Bool) (j : Fin q) => a (j,b)) using 1 congr 1 funext b cases b <;> rfl lemma pairBasis_diagonal (a : Spin q → ℂ) (x : (ι → Fin q) × (κ → Fin q)) : pairAction (diagPair a) (pairBasis x) = (∏ j, a j ^ pairContent x j) • pairBasis x := by simp only [pairBasis, OrthonormalBasis.tensorProduct_apply', EuclideanSpace.basisFun_apply, pairAction_tmul, diagPair, TensorRep.diagonal_single, TensorProduct.smul_tmul, TensorProduct.tmul_smul, smul_smul] congr 1 rw [Fintype.prod_prod_type] simp only [pairContent, Fintype.prod_bool, Bool.cond_false, Bool.cond_true] rw [Finset.prod_mul_distrib, TensorRep.product_content (fun j : Fin q => a (j,true)) x.2, TensorRep.product_content (fun j : Fin q => a (j,false)) x.1] theorem intertwiner_diagonal (L : PairRaw (q := q) (ι := ι) (κ := κ) →ₗ[ℂ] Raw (q := q) (τ := τ)) (hL : ∀ g : Group q, ∀ v, L (pairRep g v) = rep g (L v)) (a : Spin q → ℂ) (v : PairRaw (q := q) (ι := ι) (κ := κ)) : L (pairAction (diagPair a) v) = TensorRep.action (Matrix.diagonal a) (L v) := by have hh := TorusNaturality.diagonal (pairBasis (q := q) (ι := ι) (κ := κ)).toBasis (EuclideanSpace.basisFun (τ → Spin q) ℂ).toBasis pairContent TensorRep.content (fun a => pairAction (diagPair a)) (fun a => TensorRep.action (Matrix.diagonal a)) pairBasis_diagonal (fun a x => by simpa only [OrthonormalBasis.coe_toBasis, EuclideanSpace.basisFun_apply, TensorRep.product_content] using (TensorRep.diagonal_single a x)) L (fun a ha v => by let g : Group q := (⟨(diagPair a).1, TensorRep.diagonal_unitary_of_star_mul _ (fun j => ha (j,false))⟩, ⟨(diagPair a).2, TensorRep.diagonal_unitary_of_star_mul _ (fun j => ha (j,true))⟩) have hh := hL g v change L (pairAction (diagPair a) v) = TensorRep.action (block (diagPair a)) (L v) at hh rwa [block_diagPair] at hh) a exact congrArg (fun f : _ →ₗ[ℂ] _ => f v) hh lemma pair_spectral (a : Mat q × Mat q) (ha : a.1.IsHermitian) (hb : a.2.IsHermitian) : ∃ (g : Group q) (d : Spin q → ℂ), a = (g.1.val,g.2.val) * diagPair d * (g.1⁻¹.val,g.2⁻¹.val) := by refine ⟨(ha.eigenvectorUnitary,hb.eigenvectorUnitary), (fun j => cond j.2 (hb.eigenvalues j.1 : ℂ) (ha.eigenvalues j.1 : ℂ)), ?_⟩ apply Prod.ext · exact ha.spectral_theorem · exact hb.spectral_theorem theorem intertwiner_hermitian (L : PairRaw (q := q) (ι := ι) (κ := κ) →ₗ[ℂ] Raw (q := q) (τ := τ)) (hL : ∀ g : Group q, ∀ v, L (pairRep g v) = rep g (L v)) (a : Mat q × Mat q) (ha : a.1.IsHermitian) (hb : a.2.IsHermitian) (v : PairRaw (q := q) (ι := ι) (κ := κ)) : L (pairAction a v) = TensorRep.action (block a) (L v) := by obtain ⟨g,d,he⟩ := pair_spectral a ha hb rw [he] simp only [map_mul, Module.End.mul_apply, block_diagPair] change L (pairRep g (pairAction (diagPair d) (pairRep g⁻¹ v))) = rep g (TensorRep.action (Matrix.diagonal d) (rep g⁻¹ (L v))) rw [hL, intertwiner_diagonal L hL, hL] end SpinAngle.SignedTensor noncomputable section open scoped InnerProductSpace namespace SpinAngle.UnitaryNaturality variable {G : Type*} [contextInstance1640_21 : Group G] variable {E F : Type*} [contextInstance1641_23 : NormedAddCommGroup E] [contextInstance1641_46 : InnerProductSpace ℂ E] [contextInstance1642_2 : NormedAddCommGroup F] [contextInstance1642_25 : InnerProductSpace ℂ F] [contextInstance1643_2 : FiniteDimensional ℂ E] [contextInstance1643_26 : FiniteDimensional ℂ F] variable (ρ : Representation ℂ G E) (σ : Representation ℂ G F) (hρ : ∀ g x y, ⟪ρ g x,ρ g y⟫_ℂ = ⟪x,y⟫_ℂ) (hσ : ∀ g x y, ⟪σ g x,σ g y⟫_ℂ = ⟪x,y⟫_ℂ) lemma inner_move {G : Type*} [contextInstance1640_21 : Group G] {E : Type*} [contextInstance1641_23 : NormedAddCommGroup E] [contextInstance1641_46 : InnerProductSpace ℂ E] [FiniteDimensional ℂ E] (ρ : Representation ℂ G E) (hρ : ∀ g x y, ⟪ρ g x,ρ g y⟫_ℂ = ⟪x,y⟫_ℂ) (g : G) (x y : E) : ⟪x,ρ g y⟫_ℂ = ⟪ρ g⁻¹ x,y⟫_ℂ := by have h := hρ g (ρ g⁻¹ x) y simpa only [← Module.End.mul_apply, ← map_mul, mul_inv_cancel, map_one, Module.End.one_apply] using h include hρ hσ in lemma adjoint_intertwines (L : E →ₗ[ℂ] F) (hL : ∀ g x, L (ρ g x) = σ g (L x)) (g : G) (y : F) : LinearMap.adjoint L (σ g y) = ρ g (LinearMap.adjoint L y) := by apply ext_inner_left ℂ intro x rw [LinearMap.adjoint_inner_right, inner_move σ hσ, ← hL, ← LinearMap.adjoint_inner_right, ← inner_move ρ hρ] lemma adjoint_isometry (L : E →ₗᵢ[ℂ] F) (x : E) : LinearMap.adjoint L.toLinearMap (L x) = x := by apply ext_inner_left ℂ intro y rw [LinearMap.adjoint_inner_right] exact L.inner_map_map y x end SpinAngle.UnitaryNaturality namespace SpinAngle.SignedCarrier open scoped Classical TensorProduct InnerProductSpace open SpinAngle.Young SpinAngle.Specht PolynomialCarrier Module SignedTensor variable (a b : YoungDiagram) (q : ℕ) (ha : a.colLen 0 ≤ q) (hb : b.colLen 0 ≤ q) abbrev Carrier := PolynomialCarrier.Carrier a q ha ⊗[ℂ] PolynomialCarrier.Carrier b q hb instance carrierNormed : NormedAddCommGroup (Carrier a b q ha hb) := TensorProduct.instNormedAddCommGroup (𝕜 := ℂ) (E := PolynomialCarrier.Carrier a q ha) (F := PolynomialCarrier.Carrier b q hb) instance carrierInner : InnerProductSpace ℂ (Carrier a b q ha hb) := TensorProduct.instInnerProductSpace (𝕜 := ℂ) (E := PolynomialCarrier.Carrier a q ha) (F := PolynomialCarrier.Carrier b q hb) instance carrierFinite : FiniteDimensional ℂ (Carrier a b q ha hb) := inferInstanceAs (FiniteDimensional ℂ (PolynomialCarrier.Carrier a q ha ⊗[ℂ] PolynomialCarrier.Carrier b q hb)) abbrev rep : Representation ℂ (Group q) (Carrier a b q ha hb) := ExternalTensor.rep (E := PolynomialCarrier.Carrier a q ha) (F := PolynomialCarrier.Carrier b q hb) (ordinary a q ha).toRepresentation (ordinary b q hb).toRepresentation instance irreducible : (rep a b q ha hb).IsIrreducible := by let := ordinary_irreducible a q ha let := ordinary_irreducible b q hb exact ExternalTensor.irreducible _ _ (ordinary_rep_unitary a q ha) (ordinary_rep_unitary b q hb) def inclusion : Carrier a b q ha hb →ₗᵢ[ℂ] PairRaw (q := q) (ι := Cells a) (κ := Cells b) := TensorProduct.mapIsometry (E := PolynomialCarrier.Carrier a q ha) (F := PolynomialCarrier.Carrier b q hb) (PolynomialCarrier.inclusion a q ha) (PolynomialCarrier.inclusion b q hb) lemma inclusion_equivariant (g : Group q) (v : Carrier a b q ha hb) : inclusion a b q ha hb (rep a b q ha hb g v) = pairRep g (inclusion a b q ha hb v) := by induction v using TensorProduct.inductionOn with | tmul x y => rfl | add x y hx hy => simp [hx,hy] lemma rep_unitary (g : Group q) (v w : Carrier a b q ha hb) : ⟪rep a b q ha hb g v,rep a b q ha hb g w⟫_ℂ = ⟪v,w⟫_ℂ := by rw [← (inclusion a b q ha hb).inner_map_map, ← (inclusion a b q ha hb).inner_map_map v w, inclusion_equivariant, inclusion_equivariant] exact SignedTensor.pairRep_unitary g _ _ def hermitianAction (A : Mat q × Mat q) (hA : A.1.IsHermitian) (hB : A.2.IsHermitian) : Carrier a b q ha hb →ₗ[ℂ] Carrier a b q ha hb := TensorProduct.map (PolynomialCarrier.hermitianAction a q ha A.1 hA) (PolynomialCarrier.hermitianAction b q hb A.2 hB) lemma inclusion_hermitianAction (A : Mat q × Mat q) (hA : A.1.IsHermitian) (hB : A.2.IsHermitian) (v : Carrier a b q ha hb) : inclusion a b q ha hb (hermitianAction a b q ha hb A hA hB v) = pairAction A (inclusion a b q ha hb v) := by induction v using TensorProduct.inductionOn with | tmul x y => simp only [inclusion, hermitianAction, TensorProduct.mapIsometry_apply, TensorProduct.map_tmul, pairAction_tmul] change PolynomialCarrier.inclusion a q ha (PolynomialCarrier.hermitianAction a q ha A.1 hA x) ⊗ₜ[ℂ] PolynomialCarrier.inclusion b q hb (PolynomialCarrier.hermitianAction b q hb A.2 hB y) = _ rw [PolynomialCarrier.inclusion_hermitianAction, PolynomialCarrier.inclusion_hermitianAction] rfl | add x y hx hy => simp [hx,hy] variable {τ : Type*} [contextInstance1741_21 : Fintype τ] [contextInstance1741_33 : DecidableEq τ] theorem intertwiner_hermitian (L : (rep a b q ha hb).IntertwiningMap (SignedTensor.rep (q := q) (τ := τ))) (A : Mat q × Mat q) (hA : A.1.IsHermitian) (hB : A.2.IsHermitian) (v : Carrier a b q ha hb) : L (hermitianAction a b q ha hb A hA hB v) = TensorRep.action (block A) (L v) := by let J := inclusion a b q ha hb let L' := L.toLinearMap.comp (LinearMap.adjoint J.toLinearMap) have hequiv (g : Group q) (x : PairRaw (q := q) (ι := Cells a) (κ := Cells b)) : L' (pairRep g x) = SignedTensor.rep g (L' x) := by dsimp [L'] rw [UnitaryNaturality.adjoint_intertwines (rep a b q ha hb) pairRep (rep_unitary a b q ha hb) SignedTensor.pairRep_unitary J.toLinearMap (inclusion_equivariant a b q ha hb), L.isIntertwining] have h := SignedTensor.intertwiner_hermitian L' hequiv A hA hB (J v) have hj (x : Carrier a b q ha hb) : L' (J x) = L x := by dsimp [L'] rw [UnitaryNaturality.adjoint_isometry] rw [← inclusion_hermitianAction a b q ha hb A hA hB v, hj, hj] at h exact h end SpinAngle.SignedCarrier namespace SpinAngle.TensorRep variable {ι n : Type*} [contextInstance1770_23 : Fintype ι] [contextInstance1770_35 : DecidableEq ι] [contextInstance1770_51 : Fintype n] [contextInstance1770_63 : DecidableEq n] lemma continuous_action (v : EuclideanSpace ℂ (ι → n)) : Continuous (fun A : Matrix n n ℂ => action A v) := by change Continuous (fun A : Matrix n n ℂ => WithLp.toLp 2 (fun x : ι → n => ∑ y, (∏ i, A (x i) (y i)) * v y)) apply (PiLp.continuous_toLp 2 (fun _ : ι → n => ℂ)).comp fun_prop lemma continuous_rep : Continuous (fun g : Matrix.unitaryGroup n ℂ => (rep (ι := ι) g).toContinuousLinearMap) := by apply continuous_clm_apply.mpr intro v exact (continuous_action v).comp continuous_subtype_val end SpinAngle.TensorRep namespace SpinAngle.PolynomialCarrier open SpinAngle.Young SpinAngle.Specht variable (D : YoungDiagram) (q : ℕ) (hq : D.colLen 0 ≤ q) lemma continuous_rep : Continuous (fun g : Matrix.unitaryGroup (Fin q) ℂ => ((ordinary D q hq).toRepresentation g).toContinuousLinearMap) := by let : InnerProductSpace ℂ (ordinary D q hq).toSubmodule := carrierInner D q hq apply (continuous_clm_apply (𝕜 := ℂ) (E := Carrier D q hq) (F := Carrier D q hq)).mpr intro v apply (inclusion D q hq).comp_continuous_iff.mp change Continuous (fun g : Matrix.unitaryGroup (Fin q) ℂ => TensorRep.rep g (inclusion D q hq v)) exact continuous_clm_apply.mp (TensorRep.continuous_rep (ι := Cells D) (n := Fin q)) _ end SpinAngle.PolynomialCarrier namespace SpinAngle.ExternalTensor open scoped TensorProduct variable {G H : Type*} [contextInstance1802_23 : Group G] [contextInstance1802_33 : Group H] [contextInstance1802_43 : TopologicalSpace G] [contextInstance1802_64 : TopologicalSpace H] variable {E F : Type*} [contextInstance1803_23 : NormedAddCommGroup E] [contextInstance1803_46 : InnerProductSpace ℂ E] [contextInstance1804_2 : NormedAddCommGroup F] [contextInstance1804_25 : InnerProductSpace ℂ F] [contextInstance1805_2 : FiniteDimensional ℂ E] [contextInstance1805_26 : FiniteDimensional ℂ F] lemma continuous_rep (ρ : Representation ℂ G E) (σ : Representation ℂ H F) (hρ : Continuous (fun g => (ρ g).toContinuousLinearMap)) (hσ : Continuous (fun h => (σ h).toContinuousLinearMap)) : Continuous (fun g : G × H => (rep ρ σ g).toContinuousLinearMap) := by apply continuous_clm_apply.mpr intro v induction v using TensorProduct.inductionOn with | tmul x y => change Continuous (fun g : G × H => ρ g.1 x ⊗ₜ[ℂ] σ g.2 y) exact TensorProduct.continuous_tmul.comp (((continuous_clm_apply.mp hρ x).comp continuous_fst).prodMk ((continuous_clm_apply.mp hσ y).comp continuous_snd)) | add x y hx hy => have hh : Continuous (fun g : G × H => (rep ρ σ g).toContinuousLinearMap x + (rep ρ σ g).toContinuousLinearMap y) := hx.add hy convert hh using 1 funext g exact map_add _ x y end SpinAngle.ExternalTensor namespace SpinAngle.SignedTensor variable {q : ℕ} {τ : Type*} [contextInstance1828_29 : Fintype τ] [contextInstance1828_41 : DecidableEq τ] lemma continuous_block : Continuous (block (q := q)) := by apply continuous_pi intro i apply continuous_pi intro j change Continuous (fun A : Mat q × Mat q => if i.2 = j.2 then (cond i.2 A.2 A.1) i.1 j.1 else 0) split_ifs · cases i.2 <;> simp only [Bool.cond_false, Bool.cond_true] <;> fun_prop · fun_prop lemma continuous_blockGroup : Continuous (blockGroup (q := q)) := by apply Continuous.subtype_mk exact continuous_block.comp ((continuous_subtype_val.comp continuous_fst).prodMk (continuous_subtype_val.comp continuous_snd)) lemma continuous_rep : Continuous (fun g : Group q => (rep (τ := τ) g).toContinuousLinearMap) := TensorRep.continuous_rep.comp continuous_blockGroup end SpinAngle.SignedTensor namespace SpinAngle.SignedCarrier open SpinAngle.Young SpinAngle.Specht SignedTensor variable (a b : YoungDiagram) (q : ℕ) (ha : a.colLen 0 ≤ q) (hb : b.colLen 0 ≤ q) lemma continuous_rep : Continuous (fun g : Group q => (rep a b q ha hb g).toContinuousLinearMap) := ExternalTensor.continuous_rep _ _ (PolynomialCarrier.continuous_rep a q ha) (PolynomialCarrier.continuous_rep b q hb) end SpinAngle.SignedCarrier namespace SpinAngle.TensorRep open scoped Matrix.Norms.L2Operator MatrixOrder ComplexOrder variable {ι n : Type*} [contextInstance1859_23 : Fintype ι] [contextInstance1859_35 : DecidableEq ι] [contextInstance1859_51 : Fintype n] [contextInstance1859_63 : DecidableEq n] lemma tensor_posSemidef (a : ι → Matrix n n ℂ) (ha : ∀ i, (a i).PosSemidef) : (tensor a).PosSemidef := by let r : ι → Matrix n n ℂ := fun i => CFC.sqrt (a i) have he : tensor a = star (tensor r) * tensor r := by rw [← tensor_star, ← tensor_mul] congr 1 funext i dsimp [r] rw [(IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg (a i))).star_eq, CFC.sqrt_mul_sqrt_self (a i) (ha i).nonneg] rw [he] exact Matrix.posSemidef_conjTranspose_mul_self _ lemma action_positive (a : Matrix n n ℂ) (ha : a.PosSemidef) : LinearMap.IsPositive (𝕜 := ℂ) (E := EuclideanSpace ℂ (ι → n)) (action a) := Matrix.isPositive_toEuclideanLin_iff.mpr (tensor_posSemidef (fun _ : ι => a) (fun _ => ha)) end SpinAngle.TensorRep namespace SpinAngle.PolynomialCarrier open scoped Classical InnerProductSpace open SpinAngle.Young SpinAngle.Specht variable (D : YoungDiagram) (q : ℕ) (hq : D.colLen 0 ≤ q) lemma hermitianAction_positive (a : Matrix (Fin q) (Fin q) ℂ) (ha : a.PosSemidef) : LinearMap.IsPositive (𝕜 := ℂ) (E := Carrier D q hq) (hermitianAction D q hq a ha.isHermitian) := by refine ⟨hermitianAction_symmetric D q hq a ha.isHermitian, ?_⟩ intro x rw [← (inclusion D q hq).inner_map_map, inclusion_hermitianAction] exact (TensorRep.action_positive a ha).2 (inclusion D q hq x) def highestVector : Carrier D q hq := ⟨⟨highest D q hq,highest_mem D q hq⟩, self_mem_cyclic _ _⟩ lemma highestVector_ne_zero : highestVector D q hq ≠ 0 := by intro h have he := congrArg (inclusion D q hq) h exact highest_ne_zero D q hq (by simpa [highestVector] using he) lemma inclusion_linear_hermitianAction (A : Matrix (Fin q) (Fin q) ℂ) (hA : A.IsHermitian) (v : Carrier D q hq) : (inclusion D q hq).toLinearMap (hermitianAction D q hq A hA v) = TensorRep.action A ((inclusion D q hq).toLinearMap v) := inclusion_hermitianAction D q hq A hA v lemma highestVector_diagonal (t : Fin q → ℝ) : hermitianAction D q hq (Matrix.diagonal (fun i => (t i : ℂ))) (Matrix.isHermitian_diagonal_iff.mpr (fun i => by simp [IsSelfAdjoint])) (highestVector D q hq) = (highestMonomial D q hq t : ℂ) • highestVector D q hq := by have hinj : Function.Injective (inclusion D q hq).toLinearMap := by intro x y h exact Subtype.ext (Subtype.ext h) apply hinj rw [inclusion_linear_hermitianAction, map_smul] change TensorRep.action (Matrix.diagonal (fun i => (t i : ℂ))) (highest D q hq) = (highestMonomial D q hq t : ℂ) • highest D q hq rw [highest_diagonal] congr 1 exact (map_prod Complex.ofRealHom _ _).symm lemma card_rowColor (j : Fin q) : (Finset.univ.filter fun i : Cells D => rowColor D q hq i = j).card = D.rowLen j := by rw [← Fintype.card_subtype] calc Fintype.card {i : Cells D // rowColor D q hq i = j} = Fintype.card {i : Cells D // i.val.1 = j.val} := Fintype.card_congr (Equiv.subtypeEquivRight (fun i => Fin.ext_iff)) _ = D.rowLen j := card_row_subtype D j lemma highestMonomial_eq_rowLen (t : Fin q → ℝ) : highestMonomial D q hq t = ∏ j : Fin q, t j ^ D.rowLen j := by unfold highestMonomial rw [← Finset.prod_fiberwise' Finset.univ (rowColor D q hq) t] simp only [Finset.prod_const, card_rowColor] include hq in lemma sum_rowLen_eq_card : ∑ j : Fin q, D.rowLen j = Fintype.card (Cells D) := by have h := Finset.sum_fiberwise Finset.univ (rowColor D q hq) (fun _ : Cells D => (1 : ℕ)) simp only [Finset.sum_const, nsmul_eq_mul, mul_one, card_rowColor, Finset.card_univ] at h exact_mod_cast h end SpinAngle.PolynomialCarrier noncomputable section open scoped TensorProduct InnerProductSpace ComplexOrder namespace SpinAngle.TensorPositive variable {E F : Type*} [contextInstance1947_23 : NormedAddCommGroup E] [contextInstance1947_46 : InnerProductSpace ℂ E] [contextInstance1948_2 : NormedAddCommGroup F] [contextInstance1948_25 : InnerProductSpace ℂ F] [contextInstance1949_2 : FiniteDimensional ℂ E] [contextInstance1949_26 : FiniteDimensional ℂ F] lemma mapL_nonneg (A : E →L[ℂ] E) (B : F →L[ℂ] F) (hA : 0 ≤ A) (hB : 0 ≤ B) : 0 ≤ TensorProduct.mapL A B := by let a := CFC.sqrt A let b := CFC.sqrt B have ha : star a = a := (IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg A)).star_eq have hb : star b = b := (IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg B)).star_eq have hstar : star (TensorProduct.mapL a b) = TensorProduct.mapL a b := by change ContinuousLinearMap.adjoint (TensorProduct.mapL a b) = _ rw [TensorProduct.adjoint_mapL] change TensorProduct.mapL (star a) (star b) = _ rw [ha,hb] have he : star (TensorProduct.mapL a b) * TensorProduct.mapL a b = TensorProduct.mapL A B := by rw [hstar, ← TensorProduct.mapL_mul] dsimp [a,b] rw [CFC.sqrt_mul_sqrt_self A hA, CFC.sqrt_mul_sqrt_self B hB] rw [← he] exact star_mul_self_nonneg _ lemma map_positive (A : E →ₗ[ℂ] E) (B : F →ₗ[ℂ] F) (hA : A.IsPositive) (hB : B.IsPositive) : (TensorProduct.map A B).IsPositive := by have ha : 0 ≤ A.toContinuousLinearMap := (ContinuousLinearMap.nonneg_iff_isPositive).mpr hA have hb : 0 ≤ B.toContinuousLinearMap := (ContinuousLinearMap.nonneg_iff_isPositive).mpr hB have h := (ContinuousLinearMap.nonneg_iff_isPositive).mp (mapL_nonneg _ _ ha hb) change ((TensorProduct.mapL A.toContinuousLinearMap B.toContinuousLinearMap).toLinearMap).IsPositive at h simpa only [TensorProduct.toLinearMap_mapL, LinearMap.coe_toContinuousLinearMap] using h lemma map_norm_bound (A : E →ₗ[ℂ] E) (B : F →ₗ[ℂ] F) (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hA : ∀ x, ‖A x‖ ≤ a * ‖x‖) (hB : ∀ y, ‖B y‖ ≤ b * ‖y‖) (v : E ⊗[ℂ] F) : ‖TensorProduct.map A B v‖ ≤ (a*b)*‖v‖ := by have ha' : ‖A.toContinuousLinearMap‖ ≤ a := ContinuousLinearMap.opNorm_le_bound _ ha hA have hb' : ‖B.toContinuousLinearMap‖ ≤ b := ContinuousLinearMap.opNorm_le_bound _ hb hB calc ‖TensorProduct.map A B v‖ ≤ ‖TensorProduct.mapL A.toContinuousLinearMap B.toContinuousLinearMap‖ * ‖v‖ := by simpa only [TensorProduct.mapL_apply, LinearMap.coe_toContinuousLinearMap] using (ContinuousLinearMap.le_opNorm (TensorProduct.mapL A.toContinuousLinearMap B.toContinuousLinearMap) v) _ ≤ (a*b)*‖v‖ := mul_le_mul_of_nonneg_right ((TensorProduct.norm_mapL_le _ _).trans (mul_le_mul ha' hb' (norm_nonneg _) ha)) (norm_nonneg _) end SpinAngle.TensorPositive noncomputable section open scoped BigOperators namespace SpinAngle.EntropyMonomial variable {ι : Type*} [contextInstance2001_21 : Fintype ι] def entropy (n : ι → ℕ) : ℝ := (∑ i, n i : ℕ) * Real.log (∑ i, n i : ℕ) - ∑ i, (n i : ℝ) * Real.log (n i) lemma sum_log_ratio (n : ι → ℕ) (p : ℕ) (hp : 0 < p) (hn : ∑ i, n i = p) : ∑ i, (n i : ℝ) * Real.log ((n i : ℝ) / p) = -entropy n := by have hp0 : (p : ℝ) ≠ 0 := by exact_mod_cast hp.ne' have he (i : ι) : (n i : ℝ) * Real.log ((n i : ℝ) / p) = (n i : ℝ) * Real.log (n i) - (n i : ℝ) * Real.log p := by by_cases hi : n i = 0 · simp [hi] · rw [Real.log_div (by exact_mod_cast hi) hp0, mul_sub] simp only [he, Finset.sum_sub_distrib, ← Finset.sum_mul] have hn' : ∑ i, (n i : ℝ) = p := by exact_mod_cast hn rw [hn'] simp only [entropy, hn] ring lemma power_le_exponential (k p : ℕ) (hp : 0 < p) (t : ℝ) (ht : 0 ≤ t) : t ^ k ≤ Real.exp ((k : ℝ) * Real.log ((k : ℝ) / p) + p * t - k) := by by_cases hk : k = 0 · subst k simpa using (Real.one_le_exp_iff.mpr (mul_nonneg (Nat.cast_nonneg p) ht)) by_cases ht0 : t = 0 · simp only [ht0, zero_pow hk] exact (Real.exp_pos _).le have hk' : 0 < (k : ℝ) := by exact_mod_cast Nat.pos_of_ne_zero hk have hp' : 0 < (p : ℝ) := by exact_mod_cast hp have ht' : 0 < t := lt_of_le_of_ne ht (Ne.symm ht0) have hr : 0 < (k : ℝ) / p := div_pos hk' hp' have hlog := Real.log_le_sub_one_of_pos (div_pos ht' hr) rw [Real.log_div ht0 hr.ne'] at hlog have hmul := mul_le_mul_of_nonneg_left hlog hk'.le have hc : (k : ℝ) * (t / ((k : ℝ) / p) - 1) = p * t - k := by field_simp rw [hc] at hmul calc t ^ k = Real.exp ((k : ℝ) * Real.log t) := by rw [Real.exp_nat_mul, Real.exp_log ht'] _ ≤ _ := Real.exp_le_exp.mpr (by nlinarith) theorem prod_pow_le (n : ι → ℕ) (t : ι → ℝ) (ht : ∀ i, 0 ≤ t i) (hsum : ∑ i, t i ≤ 1) : ∏ i, t i ^ n i ≤ Real.exp (-entropy n) := by classical let p : ℕ := ∑ i, n i by_cases hp : p = 0 · have hn : ∀ i, n i = 0 := by intro i have hle := Finset.single_le_sum (fun j (_ : j ∈ Finset.univ) => Nat.zero_le (n j)) (Finset.mem_univ i) change n i ≤ p at hle omega simp [entropy, hn] have hp' : 0 < p := Nat.pos_of_ne_zero hp calc ∏ i, t i ^ n i ≤ ∏ i, Real.exp ((n i : ℝ) * Real.log ((n i : ℝ) / p) + p * t i - n i) := Finset.prod_le_prod₀ (fun i _ => pow_nonneg (ht i) _) (fun i _ => power_le_exponential _ p hp' _ (ht i)) _ = Real.exp (∑ i, ((n i : ℝ) * Real.log ((n i : ℝ) / p) + p * t i - n i)) := (Real.exp_sum _ _).symm _ ≤ Real.exp (-entropy n) := by apply Real.exp_le_exp.mpr simp only [Finset.sum_sub_distrib, Finset.sum_add_distrib, ← Finset.mul_sum] rw [sum_log_ratio n p hp' rfl] have hn : ∑ i, (n i : ℝ) = p := by simp [p] rw [hn] have h := mul_le_mul_of_nonneg_left hsum (Nat.cast_nonneg p : (0 : ℝ) ≤ p) nlinarith lemma optimal_prod (n : ι → ℕ) (p : ℕ) (hp : 0 < p) (hn : ∑ i, n i = p) : ∏ i, ((n i : ℝ) / p) ^ n i = Real.exp (-entropy n) := by rw [← sum_log_ratio n p hp hn, Real.exp_sum] apply Finset.prod_congr rfl intro i _ by_cases hi : n i = 0 · simp [hi] have hpos : 0 < (n i : ℝ) / p := div_pos (by exact_mod_cast Nat.pos_of_ne_zero hi) (by exact_mod_cast hp) rw [Real.exp_nat_mul, Real.exp_log hpos] end SpinAngle.EntropyMonomial noncomputable section open scoped InnerProductSpace ComplexOrder BigOperators namespace SpinAngle.PositiveTrace variable {E : Type*} [contextInstance2090_21 : NormedAddCommGroup E] [contextInstance2090_44 : InnerProductSpace ℂ E] [contextInstance2091_2 : FiniteDimensional ℂ E] lemma eigenvalues_nonneg (A : E →ₗ[ℂ] E) (hA : A.IsPositive) (i : Fin (Module.finrank ℂ E)) : 0 ≤ hA.isSymmetric.eigenvalues rfl i := by have h := hA.inner_nonneg_right (hA.isSymmetric.eigenvectorBasis rfl i) rw [hA.isSymmetric.apply_eigenvectorBasis rfl i, inner_smul_right, (hA.isSymmetric.eigenvectorBasis rfl).inner_eq_one] at h simpa only [mul_one, RCLike.ofReal_nonneg] using h theorem eigenvalue_le_trace (A : E →ₗ[ℂ] E) (hA : A.IsPositive) (v : E) (hv : v ≠ 0) (t : ℝ) (he : A v = (t : ℂ) • v) : t ≤ (LinearMap.trace ℂ E A).re := by classical let b := hA.isSymmetric.eigenvectorBasis rfl have hcoord : ∃ i, b.repr v i ≠ 0 := by by_contra! hh apply hv apply b.repr.injective ext i simpa using hh i obtain ⟨i,hi⟩ := hcoord have h := congrArg (fun x : E => b.repr x i) he rw [hA.isSymmetric.eigenvectorBasis_apply_self_apply, map_smul, PiLp.smul_apply, smul_eq_mul] at h have ht : hA.isSymmetric.eigenvalues rfl i = t := by exact Complex.ofReal_injective (mul_right_cancel₀ hi h) change t ≤ RCLike.re (LinearMap.trace ℂ E A) rw [hA.isSymmetric.re_trace_eq_sum_eigenvalues rfl, ← ht] exact Finset.single_le_sum (fun j _ => eigenvalues_nonneg A hA j) (Finset.mem_univ i) end SpinAngle.PositiveTrace noncomputable section open scoped InnerProductSpace BigOperators ComplexOrder namespace SpinAngle.Isotypic variable {G : Type*} [contextInstance2131_21 : Group G] variable {E F : Type*} [contextInstance2132_23 : NormedAddCommGroup E] [contextInstance2132_46 : InnerProductSpace ℂ E] [contextInstance2133_2 : NormedAddCommGroup F] [contextInstance2133_25 : InnerProductSpace ℂ F] [contextInstance2134_2 : FiniteDimensional ℂ E] [contextInstance2134_26 : FiniteDimensional ℂ F] variable (ρ : Representation ℂ G E) (σ : Representation ℂ G F) def space : Subrepresentation σ where toSubmodule := Submodule.span ℂ (Set.range fun z : ρ.IntertwiningMap σ × E => z.1 z.2) apply_mem_toSubmodule := by intro g v hv induction hv using Submodule.span_induction with | mem v hv => obtain ⟨⟨f,x⟩,rfl⟩ := hv apply Submodule.subset_span refine ⟨⟨f,ρ g x⟩, ?_⟩ exact Representation.IntertwiningMap.isIntertwining ρ σ f g x | zero => simp | add x y hx hy ihx ihy => simpa only [map_add] using Submodule.add_mem _ ihx ihy | smul a x hx ih => simpa only [map_smul] using Submodule.smul_mem _ a ih def projection : F →L[ℂ] F := (space ρ σ).toSubmodule.starProjection lemma projection_mem {G : Type*} [contextInstance2131_21 : Group G] {E : Type*} {F : Type*} [contextInstance2132_23 : NormedAddCommGroup E] [contextInstance2132_46 : InnerProductSpace ℂ E] [contextInstance2133_2 : NormedAddCommGroup F] [contextInstance2133_25 : InnerProductSpace ℂ F] [FiniteDimensional ℂ E] [contextInstance2134_26 : FiniteDimensional ℂ F] (ρ : Representation ℂ G E) (σ : Representation ℂ G F) (v : F) : projection ρ σ v ∈ space ρ σ := (space ρ σ).toSubmodule.starProjection_apply_mem v lemma projection_norm_apply_le {G : Type*} [contextInstance2131_21 : Group G] {E : Type*} {F : Type*} [contextInstance2132_23 : NormedAddCommGroup E] [contextInstance2132_46 : InnerProductSpace ℂ E] [contextInstance2133_2 : NormedAddCommGroup F] [contextInstance2133_25 : InnerProductSpace ℂ F] [FiniteDimensional ℂ E] [contextInstance2134_26 : FiniteDimensional ℂ F] (ρ : Representation ℂ G E) (σ : Representation ℂ G F) (v : F) : ‖projection ρ σ v‖ ≤ ‖v‖ := (space ρ σ).toSubmodule.norm_starProjection_apply_le v lemma map_mem {G : Type*} [contextInstance2131_21 : Group G] {E : Type*} {F : Type*} [contextInstance2132_23 : NormedAddCommGroup E] [contextInstance2132_46 : InnerProductSpace ℂ E] [contextInstance2133_2 : NormedAddCommGroup F] [contextInstance2133_25 : InnerProductSpace ℂ F] [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] (ρ : Representation ℂ G E) (σ : Representation ℂ G F) (f : ρ.IntertwiningMap σ) (x : E) : f x ∈ space ρ σ := Submodule.subset_span ⟨⟨f,x⟩,rfl⟩ lemma endomorphism_mem (d : σ.IntertwiningMap σ) (v : F) (hv : v ∈ space ρ σ) : d v ∈ space ρ σ := by induction hv using Submodule.span_induction with | mem v hv => obtain ⟨⟨f,x⟩,rfl⟩ := hv exact map_mem ρ σ (d.comp f) x | zero => simp only [map_zero]; exact (space ρ σ).toSubmodule.zero_mem | add x y hx hy ihx ihy => rw [map_add]; exact (space ρ σ).toSubmodule.add_mem ihx ihy | smul a x hx ih => rw [map_smul]; exact (space ρ σ).toSubmodule.smul_mem a ih lemma natural_mem (a : E →ₗ[ℂ] E) (A : F →ₗ[ℂ] F) (h : ∀ f : ρ.IntertwiningMap σ, ∀ x, A (f x) = f (a x)) (v : F) (hv : v ∈ space ρ σ) : A v ∈ space ρ σ := by induction hv using Submodule.span_induction with | mem v hv => obtain ⟨⟨f,x⟩,rfl⟩ := hv rw [h] exact map_mem ρ σ f _ | zero => simp only [map_zero]; exact (space ρ σ).toSubmodule.zero_mem | add x y hx hy ihx ihy => rw [map_add]; exact (space ρ σ).toSubmodule.add_mem ihx ihy | smul z x hx ih => rw [map_smul]; exact (space ρ σ).toSubmodule.smul_mem z ih lemma scalar_on_space {G : Type*} [contextInstance2131_21 : Group G] {E : Type*} {F : Type*} [contextInstance2132_23 : NormedAddCommGroup E] [contextInstance2132_46 : InnerProductSpace ℂ E] [contextInstance2133_2 : NormedAddCommGroup F] [contextInstance2133_25 : InnerProductSpace ℂ F] [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] (ρ : Representation ℂ G E) (σ : Representation ℂ G F) (A : F →ₗ[ℂ] F) (c : ℂ) (h : ∀ f : ρ.IntertwiningMap σ, ∀ x, A (f x) = c • f x) (v : F) (hv : v ∈ space ρ σ) : A v = c • v := by induction hv using Submodule.span_induction with | mem v hv => obtain ⟨⟨f,x⟩,rfl⟩ := hv; exact h f x | zero => simp | add x y hx hy ihx ihy => simp [ihx, ihy, smul_add] | smul z x hx ih => simp [ih, smul_comm c z] lemma exists_nonzero_adjoint (v : F) (hv : v ∈ space ρ σ) (hv0 : v ≠ 0) : ∃ f : ρ.IntertwiningMap σ, LinearMap.adjoint f.toLinearMap v ≠ 0 := by by_contra! hn have horth (w : F) (hw : w ∈ space ρ σ) : ⟪w,v⟫_ℂ = 0 := by induction hw using Submodule.span_induction with | mem w hw => obtain ⟨⟨f,x⟩,rfl⟩ := hw change ⟪f.toLinearMap x,v⟫_ℂ = 0 rw [← LinearMap.adjoint_inner_right, hn] simp | zero => simp | add x y hx hy ihx ihy => simp [ihx, ihy] | smul z x hx ih => simp [ih] exact hv0 (inner_self_eq_zero.mp (horth v hv)) lemma transfer_eigenvalue_bound (a : E →ₗ[ℂ] E) (A : F →ₗ[ℂ] F) (ha : a.IsSymmetric) (hA : A.IsSymmetric) (hnat : ∀ f : ρ.IntertwiningMap σ, ∀ x, A (f x) = f (a x)) (c : ℝ) (hc : ∀ x, ‖a x‖ ≤ c * ‖x‖) (v : F) (hv : v ∈ space ρ σ) (hv0 : v ≠ 0) (t : ℂ) (ht : A v = t • v) : ‖t‖ ≤ c := by obtain ⟨f,hf⟩ := exists_nonzero_adjoint ρ σ v hv hv0 have he : A.comp f.toLinearMap = f.toLinearMap.comp a := by ext x; exact hnat f x have hadj := congrArg LinearMap.adjoint he rw [LinearMap.adjoint_comp, LinearMap.adjoint_comp, ha.adjoint_eq, hA.adjoint_eq] at hadj have hev : a (LinearMap.adjoint f.toLinearMap v) = t • LinearMap.adjoint f.toLinearMap v := by have hh := congrArg (fun L : F →ₗ[ℂ] E => L v) hadj simpa only [LinearMap.comp_apply, ht, map_smul] using hh.symm have hh := hc (LinearMap.adjoint f.toLinearMap v) rw [hev, norm_smul] at hh exact (mul_le_mul_iff_left₀ (norm_pos_iff.mpr hf)).mp hh theorem norm_bound (a : E →ₗ[ℂ] E) (A : F →ₗ[ℂ] F) (ha : a.IsSymmetric) (hA : A.IsSymmetric) (hnat : ∀ f : ρ.IntertwiningMap σ, ∀ x, A (f x) = f (a x)) (c : ℝ) (hc0 : 0 ≤ c) (hc : ∀ x, ‖a x‖ ≤ c * ‖x‖) (v : F) (hv : v ∈ space ρ σ) : ‖A v‖ ≤ c * ‖v‖ := by classical let S := (space ρ σ).toSubmodule have hinv : ∀ w ∈ S, A w ∈ S := natural_mem ρ σ a A hnat let B : S →ₗ[ℂ] S := A.restrict hinv have hB : B.IsSymmetric := hA.restrict_invariant hinv let b := hB.eigenvectorBasis rfl have hb (i : Fin (Module.finrank ℂ S)) : ‖(hB.eigenvalues rfl i : ℂ)‖ ≤ c := by apply transfer_eigenvalue_bound ρ σ a A ha hA hnat c hc (b i).val (b i).property · intro hh have hn := b.orthonormal.ne_zero i exact hn (Subtype.ext hh) · exact congrArg Subtype.val (hB.apply_eigenvectorBasis rfl i) let w : S := ⟨v,hv⟩ change ‖B w‖ ≤ c * ‖w‖ apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hc0 (norm_nonneg _))).mp rw [← b.repr.norm_map (B w), ← b.repr.norm_map w, mul_pow, EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq, Finset.mul_sum] apply Finset.sum_le_sum intro i _ rw [hB.eigenvectorBasis_apply_self_apply, norm_mul, mul_pow] exact mul_le_mul_of_nonneg_right (pow_le_pow_left₀ (norm_nonneg _) (hb i) 2) (sq_nonneg _) lemma projection_commute_of_invariant {G : Type*} [Group G] {F : Type*} [contextInstance2133_2 : NormedAddCommGroup F] [contextInstance2133_25 : InnerProductSpace ℂ F] [contextInstance2134_26 : FiniteDimensional ℂ F] (S : Submodule ℂ F) (A : F →ₗ[ℂ] F) (hA : A.IsSymmetric) (hinv : ∀ v ∈ S, A v ∈ S) (v : F) : S.starProjection (A v) = A (S.starProjection v) := by apply S.eq_starProjection_of_mem_of_inner_eq_zero · exact hinv _ (S.starProjection_apply_mem v) · intro w hw rw [← map_sub, hA] exact S.starProjection_inner_eq_zero v _ (hinv _ hw) lemma projection_commute_endomorphism (d : σ.IntertwiningMap σ) (hd : d.toLinearMap.IsSymmetric) (v : F) : projection ρ σ (d v) = d (projection ρ σ v) := projection_commute_of_invariant (G := G) _ d.toLinearMap hd (endomorphism_mem ρ σ d) v lemma projection_commute_natural (a : E →ₗ[ℂ] E) (A : F →ₗ[ℂ] F) (hA : A.IsSymmetric) (h : ∀ f : ρ.IntertwiningMap σ, ∀ x, A (f x) = f (a x)) (v : F) : projection ρ σ (A v) = A (projection ρ σ v) := projection_commute_of_invariant (G := G) _ A hA (natural_mem ρ σ a A h) v end SpinAngle.Isotypic namespace SpinAngle.SignedTensor open scoped Matrix.Norms.L2Operator MatrixOrder ComplexOrder variable {q : ℕ} def IsEvenDensity (A : Mat q × Mat q) : Prop := A.1.PosSemidef ∧ A.2.PosSemidef ∧ Matrix.trace A.1 + Matrix.trace A.2 = 1 lemma block_posSemidef (A : Mat q × Mat q) (hA : A.1.PosSemidef) (hB : A.2.PosSemidef) : (block A).PosSemidef := by let R : Mat q × Mat q := (CFC.sqrt A.1, CFC.sqrt A.2) have he : block A = star (block R) * block R := by rw [← block_star, ← map_mul] congr 1 apply Prod.ext · change A.1 = star (CFC.sqrt A.1) * CFC.sqrt A.1 rw [(IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg A.1)).star_eq, CFC.sqrt_mul_sqrt_self A.1 hA.nonneg] · change A.2 = star (CFC.sqrt A.2) * CFC.sqrt A.2 rw [(IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg A.2)).star_eq, CFC.sqrt_mul_sqrt_self A.2 hB.nonneg] rw [he] exact Matrix.posSemidef_conjTranspose_mul_self _ end SpinAngle.SignedTensor namespace SpinAngle.SignedCarrier open scoped Classical TensorProduct BigOperators InnerProductSpace open SpinAngle.Young SpinAngle.Specht SignedTensor variable (a b : YoungDiagram) (q : ℕ) (ha : a.colLen 0 ≤ q) (hb : b.colLen 0 ≤ q) lemma hermitianAction_positive (A : Mat q × Mat q) (hA : A.1.PosSemidef) (hB : A.2.PosSemidef) : LinearMap.IsPositive (𝕜 := ℂ) (E := Carrier a b q ha hb) (hermitianAction a b q ha hb A hA.isHermitian hB.isHermitian) := TensorPositive.map_positive (E := PolynomialCarrier.Carrier a q ha) (F := PolynomialCarrier.Carrier b q hb) _ _ (PolynomialCarrier.hermitianAction_positive a q ha A.1 hA) (PolynomialCarrier.hermitianAction_positive b q hb A.2 hB) lemma hermitianAction_norm_bound (A : Mat q × Mat q) (hA : A.1.PosSemidef) (hB : A.2.PosSemidef) (v : Carrier a b q ha hb) : ‖hermitianAction a b q ha hb A hA.isHermitian hB.isHermitian v‖ ≤ (PolynomialCarrier.highestMonomial a q ha (SortedSpectrum.values A.1 hA.isHermitian) * PolynomialCarrier.highestMonomial b q hb (SortedSpectrum.values A.2 hB.isHermitian)) * ‖v‖ := by apply TensorPositive.map_norm_bound (E := PolynomialCarrier.Carrier a q ha) (F := PolynomialCarrier.Carrier b q hb) · exact Finset.prod_nonneg (fun i _ => SortedSpectrum.values_nonneg A.1 hA.isHermitian hA _) · exact Finset.prod_nonneg (fun i _ => SortedSpectrum.values_nonneg A.2 hB.isHermitian hB _) · exact PolynomialCarrier.hermitianAction_norm_bound a q ha A.1 hA · exact PolynomialCarrier.hermitianAction_norm_bound b q hb A.2 hB def parts : Fin q ⊕ Fin q → ℕ := Sum.elim (fun i => a.rowLen i) (fun i => b.rowLen i) def entropy : ℝ := EntropyMonomial.entropy (parts a b q) def dimension : ℕ := Module.finrank ℂ (Carrier a b q ha hb) include ha hb in lemma sum_parts : ∑ i, parts a b q i = Fintype.card (Cells a) + Fintype.card (Cells b) := by rw [Fintype.sum_sum_type] change (∑ i : Fin q, a.rowLen i) + (∑ i : Fin q, b.rowLen i) = _ rw [PolynomialCarrier.sum_rowLen_eq_card a q ha, PolynomialCarrier.sum_rowLen_eq_card b q hb] include ha hb in lemma entropy_eq_source : entropy a b q = (Fintype.card (Cells a) + Fintype.card (Cells b) : ℕ) * Real.log (Fintype.card (Cells a) + Fintype.card (Cells b) : ℕ) - (∑ i : Fin q, (a.rowLen i : ℝ) * Real.log (a.rowLen i)) - (∑ i : Fin q, (b.rowLen i : ℝ) * Real.log (b.rowLen i)) := by unfold entropy EntropyMonomial.entropy rw [sum_parts a b q ha hb, Fintype.sum_sum_type] simp only [parts, Sum.elim_inl, Sum.elim_inr] ring lemma hermitianAction_density_bound (A : Mat q × Mat q) (hA : IsEvenDensity A) (v : Carrier a b q ha hb) : ‖hermitianAction a b q ha hb A hA.1.isHermitian hA.2.1.isHermitian v‖ ≤ Real.exp (-entropy a b q) * ‖v‖ := by apply (hermitianAction_norm_bound a b q ha hb A hA.1 hA.2.1 v).trans apply mul_le_mul_of_nonneg_right _ (norm_nonneg _) rw [PolynomialCarrier.highestMonomial_eq_rowLen, PolynomialCarrier.highestMonomial_eq_rowLen] let t : Fin q ⊕ Fin q → ℝ := Sum.elim (SortedSpectrum.values A.1 hA.1.isHermitian) (SortedSpectrum.values A.2 hA.2.1.isHermitian) have ht (i) : 0 ≤ t i := by cases i with | inl i => exact SortedSpectrum.values_nonneg A.1 hA.1.isHermitian hA.1 i | inr i => exact SortedSpectrum.values_nonneg A.2 hA.2.1.isHermitian hA.2.1 i have hs : ∑ i, t i = 1 := by rw [Fintype.sum_sum_type] have he1 := SortedSpectrum.trace_eq_sum A.1 hA.1.isHermitian have he2 := SortedSpectrum.trace_eq_sum A.2 hA.2.1.isHermitian have h : (∑ i, (SortedSpectrum.values A.1 hA.1.isHermitian i : ℂ)) + (∑ i, (SortedSpectrum.values A.2 hA.2.1.isHermitian i : ℂ)) = 1 := (congrArg₂ (· + ·) he1.symm he2.symm).trans hA.2.2 have hh := congrArg Complex.re h simpa only [t, Sum.elim_inl, Sum.elim_inr, Complex.add_re, Complex.re_sum, Complex.ofReal_re, Complex.one_re] using hh have h := EntropyMonomial.prod_pow_le (parts a b q) t ht hs.le simpa only [Fintype.prod_sum_type, parts, t, entropy, Sum.elim_inl, Sum.elim_inr] using h end SpinAngle.SignedCarrier namespace SpinAngle.EntropyMonomial open scoped BigOperators variable {ι : Type*} [contextInstance2379_21 : Fintype ι] [contextInstance2379_33 : Nonempty ι] lemma exists_optimal (n : ι → ℕ) : ∃ t : ι → ℝ, (∀ i, 0 ≤ t i) ∧ (∑ i, t i = 1) ∧ (∏ i, t i ^ n i = Real.exp (-entropy n)) := by classical let p := ∑ i, n i by_cases hp : p = 0 · have hn : ∀ i, n i = 0 := by intro i have h := Finset.single_le_sum (fun j (_ : j ∈ Finset.univ) => Nat.zero_le (n j)) (Finset.mem_univ i) change n i ≤ p at h omega let i₀ := Classical.choice (inferInstance : Nonempty ι) refine ⟨fun i => if i = i₀ then 1 else 0, ?_, ?_, ?_⟩ · intro i; dsimp only; split_ifs <;> norm_num · simp · simp [hn, entropy] · refine ⟨fun i => (n i : ℝ) / p, fun i => div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _), ?_, ?_⟩ · rw [← Finset.sum_div] have he : ∑ i, (n i : ℝ) = p := by simp [p] rw [he, div_self (by exact_mod_cast hp)] · exact optimal_prod n p (Nat.pos_of_ne_zero hp) rfl end SpinAngle.EntropyMonomial namespace SpinAngle.SignedCarrier open scoped Classical TensorProduct BigOperators InnerProductSpace ComplexOrder open SpinAngle.Young SpinAngle.Specht SignedTensor variable (a b : YoungDiagram) (q : ℕ) (ha : a.colLen 0 ≤ q) (hb : b.colLen 0 ≤ q) variable {τ : Type*} [contextInstance2407_21 : Fintype τ] [contextInstance2407_33 : DecidableEq τ] abbrev typeProjection : Raw (q := q) (τ := τ) →L[ℂ] Raw (q := q) (τ := τ) := Isotypic.projection (rep a b q ha hb) SignedTensor.rep lemma density_on_space (A : Mat q × Mat q) (hA : IsEvenDensity A) (v : Raw (q := q) (τ := τ)) (hv : v ∈ Isotypic.space (rep a b q ha hb) (SignedTensor.rep (q := q) (τ := τ))) : ‖TensorRep.action (block A) v‖ ≤ Real.exp (-entropy a b q) * ‖v‖ := by exact Isotypic.norm_bound (E := Carrier a b q ha hb) (F := Raw (q := q) (τ := τ)) (rep a b q ha hb) (SignedTensor.rep (q := q) (τ := τ)) (hermitianAction a b q ha hb A hA.1.isHermitian hA.2.1.isHermitian) (TensorRep.action (block A)) (hermitianAction_positive a b q ha hb A hA.1 hA.2.1).isSymmetric (TensorRep.action_positive (block A) (block_posSemidef A hA.1 hA.2.1)).isSymmetric (fun L x => (intertwiner_hermitian a b q ha hb L A hA.1.isHermitian hA.2.1.isHermitian x).symm) (Real.exp (-entropy a b q)) (Real.exp_pos _).le (hermitianAction_density_bound a b q ha hb A hA) v hv lemma projected_density_bound (A : Mat q × Mat q) (hA : IsEvenDensity A) (v : Raw (q := q) (τ := τ)) : ‖TensorRep.action (block A) (typeProjection a b q ha hb v)‖ ≤ Real.exp (-entropy a b q) * ‖v‖ := by exact (density_on_space a b q ha hb A hA _ (Isotypic.projection_mem (rep a b q ha hb) (SignedTensor.rep (q := q) (τ := τ)) v)).trans (mul_le_mul_of_nonneg_left (Isotypic.projection_norm_apply_le (rep a b q ha hb) (SignedTensor.rep (q := q) (τ := τ)) v) (Real.exp_pos _).le) theorem global_density_upper (A : Mat q × Mat q) (hA : IsEvenDensity A) : ‖LinearMap.toContinuousLinearMap (TensorRep.action (ι := τ) (block A)) * typeProjection (τ := τ) a b q ha hb‖ ≤ Real.exp (-entropy a b q) := by apply ContinuousLinearMap.opNorm_le_bound _ (Real.exp_pos _).le intro v exact projected_density_bound a b q ha hb A hA v def highestVector : Carrier a b q ha hb := PolynomialCarrier.highestVector a q ha ⊗ₜ[ℂ] PolynomialCarrier.highestVector b q hb lemma highestVector_ne_zero : highestVector a b q ha hb ≠ 0 := by intro h have hn := congrArg norm h rw [highestVector, TensorProduct.norm_tmul (E := PolynomialCarrier.Carrier a q ha) (F := PolynomialCarrier.Carrier b q hb), norm_zero] at hn exact (mul_ne_zero (norm_ne_zero_iff.mpr (PolynomialCarrier.highestVector_ne_zero a q ha)) (norm_ne_zero_iff.mpr (PolynomialCarrier.highestVector_ne_zero b q hb))) hn lemma highestVector_diagonal (t : Fin q ⊕ Fin q → ℝ) : hermitianAction a b q ha hb (Matrix.diagonal (fun i => (t (.inl i) : ℂ)), Matrix.diagonal (fun i => (t (.inr i) : ℂ))) (Matrix.isHermitian_diagonal_iff.mpr (fun i => by simp [IsSelfAdjoint])) (Matrix.isHermitian_diagonal_iff.mpr (fun i => by simp [IsSelfAdjoint])) (highestVector a b q ha hb) = ((∏ i, t i ^ parts a b q i : ℝ) : ℂ) • highestVector a b q ha hb := by rw [hermitianAction, highestVector, TensorProduct.map_tmul, PolynomialCarrier.highestVector_diagonal, PolynomialCarrier.highestVector_diagonal, TensorProduct.smul_tmul_smul, PolynomialCarrier.highestMonomial_eq_rowLen, PolynomialCarrier.highestMonomial_eq_rowLen] congr 1 rw [Fintype.prod_sum_type, Complex.ofReal_mul] rfl lemma exists_density_trace_lower [Nonempty (Fin q)] : ∃ (A : Mat q × Mat q) (hA : IsEvenDensity A), Real.exp (-entropy a b q) ≤ (LinearMap.trace ℂ (Carrier a b q ha hb) (hermitianAction a b q ha hb A hA.1.isHermitian hA.2.1.isHermitian)).re := by obtain ⟨t,ht,hs,he⟩ := EntropyMonomial.exists_optimal (parts a b q) let A : Mat q × Mat q := (Matrix.diagonal (fun i => (t (.inl i) : ℂ)), Matrix.diagonal (fun i => (t (.inr i) : ℂ))) have hA : IsEvenDensity A := by refine ⟨Matrix.PosSemidef.diagonal (fun i => by simpa using ht (.inl i)), Matrix.PosSemidef.diagonal (fun i => by simpa using ht (.inr i)), ?_⟩ simp only [A, Matrix.trace_diagonal] have hh := congrArg Complex.ofReal hs simpa only [Fintype.sum_sum_type, Complex.ofReal_add, Complex.ofReal_sum, Complex.ofReal_one] using hh refine ⟨A,hA,?_⟩ apply PositiveTrace.eigenvalue_le_trace _ (hermitianAction_positive a b q ha hb A hA.1 hA.2.1) (highestVector a b q ha hb) (highestVector_ne_zero a b q ha hb) have h := highestVector_diagonal a b q ha hb t rw [he] at h exact h end SpinAngle.SignedCarrier noncomputable section open scoped BigOperators namespace SpinAngle.DenseLU abbrev Mat (n : ℕ) := Matrix (Fin n) (Fin n) ℂ def border {n : ℕ} (a : ℂ) (b c : Fin n → ℂ) (D : Mat n) : Mat (n+1) := fun i => Fin.cases (Fin.cases a b) (fun i => Fin.cases (c i) (D i)) i @[simp] lemma border_zero_zero {n : ℕ} (a : ℂ) (b c : Fin n → ℂ) (D : Mat n) : border a b c D 0 0 = a := rfl @[simp] lemma border_zero_succ {n : ℕ} (a : ℂ) (b c : Fin n → ℂ) (D : Mat n) (j : Fin n) : border a b c D 0 j.succ = b j := rfl @[simp] lemma border_succ_zero {n : ℕ} (a : ℂ) (b c : Fin n → ℂ) (D : Mat n) (i : Fin n) : border a b c D i.succ 0 = c i := rfl @[simp] lemma border_succ_succ {n : ℕ} (a : ℂ) (b c : Fin n → ℂ) (D : Mat n) (i j : Fin n) : border a b c D i.succ j.succ = D i j := rfl lemma border_eq {n : ℕ} (A : Mat (n+1)) : border (A 0 0) (fun j => A 0 j.succ) (fun i => A i.succ 0) (fun i j => A i.succ j.succ) = A := by ext i j refine Fin.cases ?_ (fun i => ?_) i <;> refine Fin.cases ?_ (fun j => ?_) j <;> rfl lemma continuous_border {X : Type*} [TopologicalSpace X] {n : ℕ} {a : X → ℂ} {b c : X → Fin n → ℂ} {D : X → Mat n} (ha : Continuous a) (hb : Continuous b) (hc : Continuous c) (hD : Continuous D) : Continuous (fun x => border (a x) (b x) (c x) (D x)) := by apply continuous_pi intro i apply continuous_pi intro j refine Fin.cases ?_ (fun i => ?_) i <;> refine Fin.cases ?_ (fun j => ?_) j · exact ha · exact (continuous_apply j).comp hb · exact (continuous_apply i).comp hc · exact (continuous_apply j).comp ((continuous_apply i).comp hD) lemma border_factor {n : ℕ} (a : ℂ) (ha : a ≠ 0) (b c : Fin n → ℂ) (L U : Mat n) : border 1 0 (fun i => c i / a) L * border a b 0 U = border a b c (L * U + (Matrix.of fun i j => c i / a * b j)) := by ext i j refine Fin.cases ?_ (fun i => ?_) i <;> refine Fin.cases ?_ (fun j => ?_) j · simp [Matrix.mul_apply, Fin.sum_univ_succ] · simp [Matrix.mul_apply, Fin.sum_univ_succ] · simp [Matrix.mul_apply, Fin.sum_univ_succ, ha] · simp [Matrix.mul_apply, Fin.sum_univ_succ, add_comm] def LowerUnipotent {n : ℕ} (L : Mat n) : Prop := L.IsLowerTriangular ∧ ∀ i, L i i = 1 lemma lower_border {n : ℕ} (c : Fin n → ℂ) (L : Mat n) (hL : LowerUnipotent L) : LowerUnipotent (border 1 0 c L) := by constructor · intro i j hij change i < j at hij induction i using Fin.cases <;> induction j using Fin.cases · simp at hij · rfl · simp at hij · exact hL.1 (by simpa using hij) · intro i refine Fin.cases ?_ (fun i => ?_) i · rfl · exact hL.2 i lemma upper_border {n : ℕ} (a : ℂ) (b : Fin n → ℂ) (U : Mat n) (hU : U.IsUpperTriangular) : (border a b 0 U).IsUpperTriangular := by intro i j hij induction i using Fin.cases <;> induction j using Fin.cases · simp at hij · simp at hij · rfl · exact hU (by simpa using hij) def bigCell (n : ℕ) : Set (Mat n) := {A | ∃ L U, LowerUnipotent L ∧ U.IsUpperTriangular ∧ A = L * U} lemma border_mem_bigCell {n : ℕ} (a : ℂ) (ha : a ≠ 0) (b c : Fin n → ℂ) (S : Mat n) (hS : S ∈ bigCell n) : border a b c (S + (Matrix.of fun i j => c i / a * b j)) ∈ bigCell (n+1) := by obtain ⟨L,U,hL,hU,rfl⟩ := hS exact ⟨border 1 0 (fun i => c i / a) L, border a b 0 U, lower_border _ _ hL, upper_border _ _ _ hU, (border_factor a ha b c L U).symm⟩ theorem dense_bigCell (n : ℕ) : Dense (bigCell n) := by induction n with | zero => intro A apply subset_closure refine ⟨1,1,?_,?_,?_⟩ · refine ⟨?_,?_⟩ <;> intro i <;> exact Fin.elim0 i · intro i; exact Fin.elim0 i · exact Subsingleton.elim _ _ | succ n ih => have hpivot (a : ℂ) (ha : a ≠ 0) (b c : Fin n → ℂ) (D : Mat n) : border a b c D ∈ closure (bigCell (n+1)) := by let X : Mat n := fun i j => c i / a * b j have hf : Continuous (fun S : Mat n => border a b c (S + X)) := continuous_border continuous_const continuous_const continuous_const (continuous_id.add continuous_const) have hh := map_mem_closure hf (ih (D-X)) (fun S hS => border_mem_bigCell a ha b c S hS) change border a b c (D - X + X) ∈ closure (bigCell (n+1)) at hh simpa only [sub_add_cancel] using hh intro A let b : Fin n → ℂ := fun j => A 0 j.succ let c : Fin n → ℂ := fun i => A i.succ 0 let D : Mat n := fun i j => A i.succ j.succ have hf : Continuous (fun a : ℂ => border a b c D) := continuous_border continuous_id continuous_const continuous_const continuous_const have hh := map_mem_closure hf ((dense_compl_singleton (0 : ℂ)) (A 0 0)) (fun a ha => hpivot a (by simpa using ha) b c D) simpa only [closure_closure, b, c, D, border_eq] using hh end SpinAngle.DenseLU namespace SpinAngle.PolynomialCarrier open scoped Classical Matrix open SpinAngle.Specht SpinAngle.Young variable (D : YoungDiagram) (q : ℕ) (hq : D.colLen 0 ≤ q) lemma action_mem_alternating (A : Matrix (Fin q) (Fin q) ℂ) (v : TensorSpace D q) (hv : v ∈ alternating D q) : TensorRep.action A v ∈ alternating D q := by intro g rw [← action_site_commute, hv g, map_smul] lemma upper_weight_support (U : Matrix (Fin q) (Fin q) ℂ) (hU : U.IsUpperTriangular) (x : Cells D → Fin q) (hx : TensorRep.action U (highest D q hq) x ≠ 0) : ∑ i, (x i).val = ∑ i : Cells D, i.val.1 := by have hl := rowRankSum_le D (fun i => (x i).val) (alternating_injective D q _ (action_mem_alternating D q U _ (highest_mem D q hq)) x hx) apply le_antisymm _ hl change (∑ y, (∏ i, U (x i) (y i)) * highest D q hq y) ≠ 0 at hx obtain ⟨y, -, hy⟩ := Finset.exists_ne_zero_of_sum_ne_zero hx have hyv : highest D q hq y ≠ 0 := (mul_ne_zero_iff.mp hy).2 have hp : ∀ i, x i ≤ y i := by intro i by_contra hi have hz : U (x i) (y i) = 0 := hU (lt_of_not_ge hi) exact (mul_ne_zero_iff.mp hy).1 (Finset.prod_eq_zero (Finset.mem_univ i) hz) rw [← weight_eigen_support D q hq _ (highest_weight_eigen D q hq) y hyv] exact Finset.sum_le_sum fun i _ => hp i lemma upper_highest (U : Matrix (Fin q) (Fin q) ℂ) (hU : U.IsUpperTriangular) : TensorRep.action U (highest D q hq) = (TensorRep.action U (highest D q hq) (rowColor D q hq)) • highest D q hq := by ext x change _ = _ * highest D q hq x by_cases hw : ∑ i, (x i).val = ∑ i : Cells D, i.val.1 · exact alternating_minimal_weight D q hq _ (action_mem_alternating D q U _ (highest_mem D q hq)) x hw · have hv0 : TensorRep.action U (highest D q hq) x = 0 := by by_contra hn exact hw (upper_weight_support D q hq U hU x hn) have he0 : highest D q hq x = 0 := by by_contra hn exact hw (weight_eigen_support D q hq _ (highest_weight_eigen D q hq) x hn) rw [hv0, he0, mul_zero] lemma continuous_action_highest : Continuous (fun A : Matrix (Fin q) (Fin q) ℂ => TensorRep.action A (highest D q hq)) := by change Continuous (fun A : Matrix (Fin q) (Fin q) ℂ => WithLp.toLp 2 (fun x : Cells D → Fin q => ∑ y, (∏ i, A (x i) (y i)) * highest D q hq y)) apply (PiLp.continuous_toLp 2 (fun _ : Cells D → Fin q => ℂ)).comp fun_prop lemma action_highest_mem_of_lower (S : Submodule ℂ (TensorSpace D q)) (hS : ∀ L : Matrix (Fin q) (Fin q) ℂ, DenseLU.LowerUnipotent L → TensorRep.action L (highest D q hq) ∈ S) (A : Matrix (Fin q) (Fin q) ℂ) : TensorRep.action A (highest D q hq) ∈ S := by have hsub : DenseLU.bigCell q ⊆ (fun B : Matrix (Fin q) (Fin q) ℂ => TensorRep.action B (highest D q hq)) ⁻¹' (S : Set (TensorSpace D q)) := by intro B hB obtain ⟨L,U,hL,hU,rfl⟩ := hB change TensorRep.action (L * U) (highest D q hq) ∈ S rw [map_mul, Module.End.mul_apply, upper_highest D q hq U hU, map_smul] exact S.smul_mem _ (hS L hL) exact closure_minimal hsub (S.closed_of_finiteDimensional.preimage (continuous_action_highest D q hq)) (DenseLU.dense_bigCell q A) end SpinAngle.PolynomialCarrier noncomputable section open scoped BigOperators namespace SpinAngle.PolyDimension open MvPolynomial Module variable (σ : Type*) [contextInstance2689_21 : Fintype σ] [contextInstance2689_33 : DecidableEq σ] (p : ℕ) def boundedExponentEquiv : {f : σ →₀ ℕ // ∀ i, f i ≤ p} ≃ (σ → Fin (p+1)) where toFun f i := ⟨f.val i, Nat.lt_succ_of_le (f.property i)⟩ invFun f := ⟨Finsupp.equivFunOnFinite.symm (fun i => (f i).val), by intro i; exact Nat.le_of_lt_succ (f i).isLt⟩ left_inv f := by apply Subtype.ext; ext i; rfl right_inv f := by funext i; apply Fin.ext; rfl theorem finrank_restrictDegree : Module.finrank ℂ (restrictDegree σ ℂ p) = (p+1) ^ Fintype.card σ := by let s : Set (σ →₀ ℕ) := {f | ∀ i, f i ≤ p} let : Fintype s := Fintype.ofEquiv (σ → Fin (p+1)) (boundedExponentEquiv σ p).symm change Module.finrank ℂ (restrictSupport ℂ s) = _ rw [Module.finrank_eq_card_basis (basisRestrictSupport ℂ s)] calc Fintype.card s = Fintype.card (σ → Fin (p+1)) := Fintype.card_congr (boundedExponentEquiv σ p) _ = _ := by simp abbrev LowerIndex (q : ℕ) := (i : Fin q) × Fin i.val lemma card_lowerIndex (q : ℕ) : Fintype.card (LowerIndex q) = q * (q-1) / 2 := by simp only [LowerIndex, Fintype.card_sigma, Fintype.card_fin] rw [Fin.sum_univ_eq_sum_range (fun i : ℕ => i) q, Finset.sum_range_id] lemma twice_card_lowerIndex (q : ℕ) : 2 * Fintype.card (LowerIndex q) = q * (q-1) := by simp only [LowerIndex, Fintype.card_sigma, Fintype.card_fin] rw [Fin.sum_univ_eq_sum_range (fun i : ℕ => i) q, mul_comm, Finset.sum_range_id_mul_two] variable {q : ℕ} def lowerEntry (i j : Fin q) : MvPolynomial (LowerIndex q) ℂ := if h : j < i then X ⟨i, ⟨j.val, h⟩⟩ else if i = j then 1 else 0 lemma lowerEntry_degree (i j : Fin q) : (lowerEntry i j).totalDegree ≤ 1 := by unfold lowerEntry split_ifs <;> simp lemma totalDegree_sum_le (σ : Type*) [Fintype σ] [DecidableEq σ] (p : ℕ) {ι : Type*} (s : Finset ι) (f : ι → MvPolynomial σ ℂ) (hf : ∀ i ∈ s, (f i).totalDegree ≤ p) : (∑ i ∈ s, f i).totalDegree ≤ p := (MvPolynomial.totalDegree_finsetSum s f).trans (Finset.sup_le hf) end SpinAngle.PolyDimension namespace SpinAngle.PolyDimension open MvPolynomial Module variable {σ X : Type*} [contextInstance2737_23 : Fintype σ] [contextInstance2737_35 : DecidableEq σ] [contextInstance2737_51 : Fintype X] (p : ℕ) def valueTranspose (P : X → restrictDegree σ ℂ p) : Module.Dual ℂ (restrictDegree σ ℂ p) →ₗ[ℂ] EuclideanSpace ℂ X where toFun φ := WithLp.toLp 2 (fun x => φ (P x)) map_add' φ ψ := by ext x; rfl map_smul' c φ := by ext x; rfl lemma value_mem_range {σ : Type*} {X : Type*} [Fintype σ] [DecidableEq σ] [Fintype X] (p : ℕ) (P : X → restrictDegree σ ℂ p) (a : σ → ℂ) : WithLp.toLp 2 (fun x => MvPolynomial.aeval a (P x).val) ∈ LinearMap.range (valueTranspose p P) := by refine ⟨(MvPolynomial.aeval a).toLinearMap.comp (restrictDegree σ ℂ p).subtype, ?_⟩ rfl lemma finrank_value_range {σ : Type*} {X : Type*} [contextInstance2737_23 : Fintype σ] [contextInstance2737_35 : DecidableEq σ] [Fintype X] (p : ℕ) (P : X → restrictDegree σ ℂ p) : Module.finrank ℂ (LinearMap.range (valueTranspose p P)) ≤ (p+1)^Fintype.card σ := by have hh := (valueTranspose p P).finrank_range_le rw [Subspace.dual_finrank_eq, finrank_restrictDegree] at hh exact hh end SpinAngle.PolyDimension namespace SpinAngle.PolynomialCarrier open scoped Classical Matrix open SpinAngle.Specht SpinAngle.Young PolyDimension MvPolynomial variable (D : YoungDiagram) (q : ℕ) (hq : D.colLen 0 ≤ q) def lowerCoordinate (x : Cells D → Fin q) : MvPolynomial (LowerIndex q) ℂ := ∑ y : Cells D → Fin q, (∏ i : Cells D, lowerEntry (x i) (y i)) * C (highest D q hq y) lemma lowerCoordinate_degree (x : Cells D → Fin q) : (lowerCoordinate D q hq x).totalDegree ≤ Fintype.card (Cells D) := by apply PolyDimension.totalDegree_sum_le intro y _ apply (MvPolynomial.totalDegree_mul _ _).trans rw [MvPolynomial.totalDegree_C, add_zero] apply (MvPolynomial.totalDegree_finsetProd _ _).trans exact (Finset.sum_le_sum fun i _ => lowerEntry_degree (x i) (y i)).trans_eq (by simp) def lowerCoordinateBounded (x : Cells D → Fin q) : restrictDegree (LowerIndex q) ℂ (Fintype.card (Cells D)) := ⟨lowerCoordinate D q hq x, restrictTotalDegree_le_restrictDegree _ _ _ ((mem_restrictTotalDegree _ _ _).mpr (lowerCoordinate_degree D q hq x))⟩ def lowerParameters (L : Matrix (Fin q) (Fin q) ℂ) (z : LowerIndex q) : ℂ := L z.1 ⟨z.2.val, z.2.isLt.trans z.1.isLt⟩ lemma eval_lowerEntry (L : Matrix (Fin q) (Fin q) ℂ) (hL : DenseLU.LowerUnipotent L) (i j : Fin q) : aeval (lowerParameters q L) (lowerEntry i j) = L i j := by unfold lowerEntry split_ifs with hlt heq · simp [aeval_X, lowerParameters] · subst j; simp [hL.2 i] · have hij : i < j := lt_of_le_of_ne (le_of_not_gt hlt) heq rw [map_zero, hL.1 hij] lemma eval_lowerCoordinate (L : Matrix (Fin q) (Fin q) ℂ) (hL : DenseLU.LowerUnipotent L) (x : Cells D → Fin q) : aeval (lowerParameters q L) (lowerCoordinate D q hq x) = TensorRep.action L (highest D q hq) x := by simp only [lowerCoordinate, map_sum, map_mul, map_prod, eval_lowerEntry q L hL] change (∑ y : Cells D → Fin q, (∏ i, L (x i) (y i)) * aeval (lowerParameters q L) (C (highest D q hq y))) = ∑ y : Cells D → Fin q, (∏ i, L (x i) (y i)) * highest D q hq y apply Finset.sum_congr rfl intro y _ simp theorem ordinary_finrank_le : Module.finrank ℂ (ordinary D q hq).toSubmodule ≤ (Fintype.card (Cells D)+1) ^ (q*(q-1)/2) := by let P := lowerCoordinateBounded D q hq let S := LinearMap.range (PolyDimension.valueTranspose (Fintype.card (Cells D)) P) have hL (L : Matrix (Fin q) (Fin q) ℂ) (hL : DenseLU.LowerUnipotent L) : TensorRep.action L (highest D q hq) ∈ S := by have hh := PolyDimension.value_mem_range (Fintype.card (Cells D)) P (lowerParameters q L) have he : WithLp.toLp 2 (fun x => aeval (lowerParameters q L) (P x).val) = TensorRep.action L (highest D q hq) := by ext x; exact eval_lowerCoordinate D q hq L hL x rw [he] at hh exact hh let inc : (ordinary D q hq).toSubmodule →ₗ[ℂ] TensorSpace D q := (alternating D q).toSubmodule.subtype.comp (ordinary D q hq).toSubmodule.subtype have hcarrier (v : (alternating D q).toSubmodule) (hv : v ∈ (ordinary D q hq).toSubmodule) : (v : TensorSpace D q) ∈ S := by change v ∈ Submodule.span ℂ (Set.range fun g => (alternating D q).toRepresentation g (⟨highest D q hq, highest_mem D q hq⟩ : (alternating D q).toSubmodule)) at hv induction hv using Submodule.span_induction with | mem v hv => obtain ⟨g,rfl⟩ := hv exact action_highest_mem_of_lower D q hq S hL g.val | zero => exact S.zero_mem | add x y hx hy ihx ihy => exact S.add_mem ihx ihy | smul a x hx ih => exact S.smul_mem a ih have hinc (v : (ordinary D q hq).toSubmodule) : inc v ∈ S := hcarrier v.val v.property let f : (ordinary D q hq).toSubmodule →ₗ[ℂ] S := LinearMap.codRestrict S inc hinc have hf : Function.Injective f := by intro x y he apply Subtype.ext apply Subtype.ext exact congrArg (fun v : S => v.val) he have hh := LinearMap.finrank_le_finrank_of_injective hf exact hh.trans (by simpa only [PolyDimension.card_lowerIndex] using PolyDimension.finrank_value_range (Fintype.card (Cells D)) P) end SpinAngle.PolynomialCarrier noncomputable section open MeasureTheory TopologicalSpace Set open scoped BigOperators namespace SpinAngle.CompactTwirl variable {G : Type*} [contextInstance2861_21 : Group G] [contextInstance2861_31 : TopologicalSpace G] [contextInstance2861_52 : IsTopologicalGroup G] [contextInstance2862_2 : CompactSpace G] [contextInstance2862_19 : MeasurableSpace G] [contextInstance2862_39 : BorelSpace G] def compactHaar : Measure G := Measure.haarMeasure (⟨⟨Set.univ, isCompact_univ⟩, by simp⟩ : PositiveCompacts G) instance : IsProbabilityMeasure (compactHaar (G := G)) where measure_univ := Measure.haarMeasure_self instance : (compactHaar (G := G)).IsMulLeftInvariant := inferInstanceAs (Measure.IsMulLeftInvariant (Measure.haarMeasure _)) lemma continuous_integrable {A : Type*} [NormedAddCommGroup A] (f : G → A) (hf : Continuous f) : Integrable f (compactHaar (G := G)) := hf.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f) variable {A : Type*} [contextInstance2877_21 : NormedRing A] [contextInstance2877_36 : NormedAlgebra ℝ A] [contextInstance2877_56 : CompleteSpace A] def conjugate (ρ : G →* A) (a : A) (g : G) : A := ρ g * a * ρ g⁻¹ def twirl (ρ : G →* A) (a : A) : A := ∫ g, conjugate ρ a g ∂compactHaar lemma continuous_conjugate {G : Type*} [contextInstance2861_21 : Group G] [contextInstance2861_31 : TopologicalSpace G] [contextInstance2861_52 : IsTopologicalGroup G] [CompactSpace G] [MeasurableSpace G] [BorelSpace G] {A : Type*} [contextInstance2877_21 : NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] (ρ : G →* A) (hρ : Continuous ρ) (a : A) : Continuous (conjugate ρ a) := (hρ.mul continuous_const).mul (hρ.comp continuous_inv) lemma integrable_conjugate (ρ : G →* A) (hρ : Continuous ρ) (a : A) : Integrable (conjugate ρ a) compactHaar := continuous_integrable _ (continuous_conjugate ρ hρ a) lemma conjugate_mul {G : Type*} [contextInstance2861_21 : Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [MeasurableSpace G] [BorelSpace G] {A : Type*} [contextInstance2877_21 : NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] (ρ : G →* A) (a : A) (g h : G) : conjugate ρ a (g * h) = ρ g * conjugate ρ a h * ρ g⁻¹ := by simp only [conjugate, mul_inv_rev, map_mul, mul_assoc] lemma twirl_commute (ρ : G →* A) (hρ : Continuous ρ) (a : A) (g : G) : Commute (ρ g) (twirl ρ a) := by have hi := integrable_conjugate ρ hρ a have hinv : (ρ g⁻¹) * ρ g = 1 := by rw [← map_mul, inv_mul_cancel, map_one] have he : ρ g * twirl ρ a * ρ g⁻¹ = twirl ρ a := by rw [twirl, ← integral_const_mul_of_integrable hi] rw [← integral_mul_const_of_integrable (hi.const_mul _)] simp_rw [← conjugate_mul] exact integral_mul_left_eq_self _ g have hc := congrArg (fun x : A => x * ρ g) he simpa only [Commute, SemiconjBy, mul_assoc, hinv, mul_one] using hc lemma twirl_trace (τ : A →L[ℝ] ℂ) (hτ : ∀ x y : A, τ (x * y) = τ (y * x)) (ρ : G →* A) (hρ : Continuous ρ) (a : A) : τ (twirl ρ a) = τ a := by rw [twirl, ← τ.integral_comp_comm (integrable_conjugate ρ hρ a)] have he (g : G) : τ (conjugate ρ a g) = τ a := by rw [conjugate, hτ, ← mul_assoc, ← map_mul, inv_mul_cancel, map_one, one_mul] simp [he, integral_const, measureReal_def] section Positivity variable [contextInstance2919_9 : PartialOrder A] [contextInstance2919_26 : IsOrderedAddMonoid A] [contextInstance2919_49 : StarRing A] [contextInstance2919_62 : StarOrderedRing A] [contextInstance2920_2 : ClosedIciTopology A] [contextInstance2920_24 : IsOrderedModule ℝ A] lemma twirl_nonneg {G : Type*} [contextInstance2861_21 : Group G] [contextInstance2861_31 : TopologicalSpace G] [contextInstance2861_52 : IsTopologicalGroup G] [contextInstance2862_2 : CompactSpace G] [contextInstance2862_19 : MeasurableSpace G] [contextInstance2862_39 : BorelSpace G] {A : Type*} [contextInstance2877_21 : NormedRing A] [contextInstance2877_36 : NormedAlgebra ℝ A] [CompleteSpace A] [contextInstance2919_9 : PartialOrder A] [contextInstance2919_26 : IsOrderedAddMonoid A] [contextInstance2919_49 : StarRing A] [contextInstance2919_62 : StarOrderedRing A] [contextInstance2920_2 : ClosedIciTopology A] [contextInstance2920_24 : IsOrderedModule ℝ A] (ρ : G →* A) (a : A) (hρ : ∀ g, ρ g⁻¹ = star (ρ g)) (ha : 0 ≤ a) : 0 ≤ twirl ρ a := by apply integral_nonneg intro g dsimp [conjugate] rw [hρ] exact star_right_conjugate_nonneg ha _ end Positivity end SpinAngle.CompactTwirl namespace SpinAngle open scoped Matrix.Norms.L2Operator lemma unitaryGroup_compactSpace (n : Type*) [Fintype n] [DecidableEq n] [Nonempty n] : CompactSpace (Matrix.unitaryGroup n ℂ) := by apply isCompact_iff_compactSpace.mp apply (isCompact_closedBall (0 : Matrix n n ℂ) 1).of_isClosed_subset isClosed_unitary intro a ha rw [Metric.mem_closedBall, dist_zero_right] exact (CStarRing.norm_coe_unitary (⟨a, ha⟩ : unitary (Matrix n n ℂ))).le end SpinAngle namespace SpinAngle.CompactTwirl variable {G : Type*} [contextInstance2949_21 : Group G] [contextInstance2949_31 : TopologicalSpace G] [contextInstance2949_52 : IsTopologicalGroup G] [contextInstance2950_2 : CompactSpace G] [contextInstance2950_19 : MeasurableSpace G] [contextInstance2950_39 : BorelSpace G] variable {E : Type*} [contextInstance2951_21 : NormedAddCommGroup E] [contextInstance2951_44 : InnerProductSpace ℂ E] [contextInstance2952_2 : FiniteDimensional ℂ E] def continuousAction (ρ : Representation ℂ G E) : G →* (E →L[ℂ] E) := (Module.End.toContinuousLinearMap (𝕜 := ℂ) E).toMonoidHom.comp ρ @[simp] lemma continuousAction_apply {G : Type*} [contextInstance2949_21 : Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [MeasurableSpace G] [BorelSpace G] {E : Type*} [contextInstance2951_21 : NormedAddCommGroup E] [contextInstance2951_44 : InnerProductSpace ℂ E] [contextInstance2952_2 : FiniteDimensional ℂ E] (ρ : Representation ℂ G E) (g : G) (v : E) : continuousAction ρ g v = ρ g v := rfl def operatorTrace : (E →L[ℂ] E) →L[ℂ] ℂ := ((LinearMap.trace ℂ E).comp (Module.End.toContinuousLinearMap (𝕜 := ℂ) E).symm.toLinearMap).toContinuousLinearMap @[simp] lemma operatorTrace_apply {G : Type*} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [MeasurableSpace G] [BorelSpace G] {E : Type*} [contextInstance2951_21 : NormedAddCommGroup E] [contextInstance2951_44 : InnerProductSpace ℂ E] [contextInstance2952_2 : FiniteDimensional ℂ E] (a : E →L[ℂ] E) : operatorTrace a = LinearMap.trace ℂ E a.toLinearMap := rfl lemma operatorTrace_mul_comm {G : Type*} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [MeasurableSpace G] [BorelSpace G] {E : Type*} [contextInstance2951_21 : NormedAddCommGroup E] [contextInstance2951_44 : InnerProductSpace ℂ E] [FiniteDimensional ℂ E] (a b : E →L[ℂ] E) : operatorTrace (a * b) = operatorTrace (b * a) := LinearMap.trace_mul_comm ℂ a.toLinearMap b.toLinearMap theorem twirl_irreducible (ρ : Representation ℂ G E) [ρ.IsIrreducible] (hρ : Continuous (continuousAction ρ)) (a : E →L[ℂ] E) : twirl (continuousAction ρ) a = (operatorTrace a / (Module.finrank ℂ E : ℂ)) • (1 : E →L[ℂ] E) := by let t := twirl (continuousAction ρ) a let f : ρ.IntertwiningMap ρ := t.toLinearMap.intertwiningMap_of_isIntertwiningMap ρ ρ (by intro g v have h := twirl_commute (continuousAction ρ) hρ a g exact (congrArg (fun L : E →L[ℂ] E => L v) h.eq).symm) obtain ⟨z, hz⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed (ρ := ρ)).surjective f have ht : t = z • (1 : E →L[ℂ] E) := by ext v exact (congrArg (fun L : ρ.IntertwiningMap ρ => L v) hz).symm have htr : operatorTrace t = operatorTrace a := twirl_trace (operatorTrace.restrictScalars ℝ) (operatorTrace_mul_comm (G := G)) (continuousAction ρ) hρ a let : Nontrivial E := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) ρ.asModule have hn : (Module.finrank ℂ E : ℂ) ≠ 0 := by exact_mod_cast Module.finrank_pos.ne' have hz' : z = operatorTrace a / (Module.finrank ℂ E : ℂ) := by apply (eq_div_iff hn).mpr rw [ht, map_smul] at htr simpa [operatorTrace_apply (G := G), LinearMap.trace_one] using htr simpa only [hz'] using ht end SpinAngle.CompactTwirl namespace SpinAngle.CompactTwirl open scoped InnerProductSpace ComplexOrder variable {G : Type*} [contextInstance3008_21 : Group G] [contextInstance3008_31 : TopologicalSpace G] [contextInstance3008_52 : IsTopologicalGroup G] [contextInstance3009_2 : CompactSpace G] [contextInstance3009_19 : MeasurableSpace G] [contextInstance3009_39 : BorelSpace G] variable {E F : Type*} [contextInstance3010_23 : NormedAddCommGroup E] [contextInstance3010_46 : InnerProductSpace ℂ E] [contextInstance3011_2 : NormedAddCommGroup F] [contextInstance3011_25 : InnerProductSpace ℂ F] [contextInstance3012_2 : FiniteDimensional ℂ E] [contextInstance3012_26 : FiniteDimensional ℂ F] lemma twirl_natural (ρ : Representation ℂ G E) (σ : Representation ℂ G F) (hρ : Continuous (continuousAction ρ)) (hσ : Continuous (continuousAction σ)) (a : E →L[ℂ] E) (A : F →L[ℂ] F) (L : ρ.IntertwiningMap σ) (hnat : ∀ x, A (L x) = L (a x)) (x : E) : twirl (continuousAction σ) A (L x) = L (twirl (continuousAction ρ) a x) := by have hi := integrable_conjugate (continuousAction ρ) hρ a have hI := integrable_conjugate (continuousAction σ) hσ A rw [twirl, ContinuousLinearMap.integral_apply hI, twirl, ContinuousLinearMap.integral_apply hi] change _ = L.toLinearMap.toContinuousLinearMap (∫ g, conjugate (continuousAction ρ) a g x ∂compactHaar) rw [← ContinuousLinearMap.integral_comp_comm _ (hi.apply_continuousLinearMap x)] apply integral_congr_ae filter_upwards [] with g change σ g (A (σ g⁻¹ (L x))) = L (ρ g (a (ρ g⁻¹ x))) rw [← L.isIntertwining, hnat, L.isIntertwining] lemma continuousAction_inverse_star {G : Type*} [contextInstance3008_21 : Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [MeasurableSpace G] [BorelSpace G] {E : Type*} [contextInstance3010_23 : NormedAddCommGroup E] [contextInstance3010_46 : InnerProductSpace ℂ E] [contextInstance3012_2 : FiniteDimensional ℂ E] (ρ : Representation ℂ G E) (hρ : ∀ g x y, ⟪ρ g x,ρ g y⟫_ℂ = ⟪x,y⟫_ℂ) (g : G) : continuousAction ρ g⁻¹ = star (continuousAction ρ g) := by ext x apply ext_inner_left ℂ intro y change ⟪y,ρ g⁻¹ x⟫_ℂ = ⟪y,(continuousAction ρ g).adjoint x⟫_ℂ rw [ContinuousLinearMap.adjoint_inner_right] simpa only [inv_inv, continuousAction_apply] using (UnitaryNaturality.inner_move ρ hρ g⁻¹ y x) end SpinAngle.CompactTwirl namespace SpinAngle.ProjectionOrder variable {A : Type*} [contextInstance3043_21 : CStarAlgebra A] [contextInstance3043_38 : PartialOrder A] [contextInstance3043_55 : StarOrderedRing A] [contextInstance3044_2 : NonnegSpectrumClass ℝ A] lemma scalar_projection_le (T P : A) (hT : 0 ≤ T) (hP : IsStarProjection P) (c : ℝ) (he : T * P = c • P) : c • P ≤ T := by have hstar := congrArg star he have hpT : P * T = c • P := by simpa only [star_mul, hP.isSelfAdjoint.star_eq, (IsSelfAdjoint.of_nonneg hT).star_eq, star_smul, star_trivial] using hstar have hc : Commute (1-P) T := by dsimp [Commute,SemiconjBy] rw [sub_mul,mul_sub,one_mul,mul_one,hpT,he] have hn := hc.mul_nonneg (sub_nonneg.mpr hP.le_one) hT rw [sub_mul,one_mul,hpT] at hn exact sub_nonneg.mp hn end SpinAngle.ProjectionOrder namespace SpinAngle.CompactTwirl open scoped InnerProductSpace ComplexOrder variable {G : Type*} [contextInstance3062_21 : Group G] [contextInstance3062_31 : TopologicalSpace G] [contextInstance3062_52 : IsTopologicalGroup G] [contextInstance3063_2 : CompactSpace G] [contextInstance3063_19 : MeasurableSpace G] [contextInstance3063_39 : BorelSpace G] variable {E F : Type*} [contextInstance3064_23 : NormedAddCommGroup E] [contextInstance3064_46 : InnerProductSpace ℂ E] [contextInstance3065_2 : NormedAddCommGroup F] [contextInstance3065_25 : InnerProductSpace ℂ F] [contextInstance3066_2 : FiniteDimensional ℂ E] [contextInstance3066_26 : FiniteDimensional ℂ F] lemma isotypic_twirl_scalar (ρ : Representation ℂ G E) [ρ.IsIrreducible] (σ : Representation ℂ G F) (hρ : Continuous (continuousAction ρ)) (hσ : Continuous (continuousAction σ)) (a : E →L[ℂ] E) (A : F →L[ℂ] F) (hnat : ∀ L : ρ.IntertwiningMap σ, ∀ x, A (L x) = L (a x)) : twirl (continuousAction σ) A * Isotypic.projection ρ σ = (operatorTrace a / (Module.finrank ℂ E : ℂ)) • Isotypic.projection ρ σ := by ext v apply Isotypic.scalar_on_space ρ σ (twirl (continuousAction σ) A).toLinearMap (operatorTrace a / (Module.finrank ℂ E : ℂ)) · intro L x change twirl (continuousAction σ) A (L x) = _ rw [twirl_natural ρ σ hρ hσ a A L (hnat L), twirl_irreducible ρ hρ a] simp only [smul_apply, one_apply_eq_self, map_smul] · exact (Isotypic.space ρ σ).toSubmodule.starProjection_apply_mem v lemma isotypic_twirl_lower (ρ : Representation ℂ G E) [ρ.IsIrreducible] (σ : Representation ℂ G F) (hρ : Continuous (continuousAction ρ)) (hσ : Continuous (continuousAction σ)) (hσunit : ∀ g x y, ⟪σ g x,σ g y⟫_ℂ = ⟪x,y⟫_ℂ) (a : E →L[ℂ] E) (A : F →L[ℂ] F) (ha : 0 ≤ a) (hA : 0 ≤ A) (hnat : ∀ L : ρ.IntertwiningMap σ, ∀ x, A (L x) = L (a x)) : ((operatorTrace a).re / (Module.finrank ℂ E : ℝ)) • Isotypic.projection ρ σ ≤ twirl (continuousAction σ) A := by apply ProjectionOrder.scalar_projection_le _ _ (twirl_nonneg _ A (continuousAction_inverse_star σ hσunit) hA) (isStarProjection_starProjection (U := (Isotypic.space ρ σ).toSubmodule)) have ht : operatorTrace a = ((operatorTrace a).re : ℂ) := Complex.eq_re_of_ofReal_le (LinearMap.IsPositive.trace_nonneg ((ContinuousLinearMap.nonneg_iff_isPositive).mp ha)) have h := isotypic_twirl_scalar ρ σ hρ hσ a A hnat rw [ht] at h convert h using 1 <;> simp only [Isotypic.projection, RCLike.real_smul_eq_coe_smul (K := ℂ)] congr 1 change (((operatorTrace a).re / (Module.finrank ℂ E : ℝ) : ℝ) : ℂ) = _ simp only [Complex.ofReal_div, Complex.ofReal_natCast] end SpinAngle.CompactTwirl noncomputable section open scoped BigOperators open Set namespace SpinAngle.CompactMixture variable {E : Type*} [contextInstance3119_21 : NormedAddCommGroup E] [contextInstance3119_44 : NormedSpace ℝ E] [contextInstance3119_62 : FiniteDimensional ℝ E] lemma convexHull_compact {s : Set E} (hs : IsCompact s) : IsCompact (convexHull ℝ s) := by classical let : CompactSpace s := isCompact_iff_compactSpace.mp hs let C (k : ℕ) := (Convexity.StdSimplex ℝ (Fin k)) × (Fin k → s) let f (k : ℕ) (v : C k) : E := ∑ i, v.1.weights i • (v.2 i).val have hf (k : ℕ) : Continuous (f k) := by change Continuous (fun v : (Convexity.StdSimplex ℝ (Fin k)) × (Fin k → s) => ∑ i, v.1.weights i • (v.2 i).val) apply continuous_finsetSum intro i hi exact ((Convexity.StdSimplex.continuous_weights_apply ℝ i).comp continuous_fst).smul (continuous_subtype_val.comp ((continuous_apply i).comp continuous_snd)) have hc : IsCompact (⋃ k : Fin (Module.finrank ℝ E + 2), Set.range (f k.val)) := isCompact_iUnion (fun k => isCompact_range (hf k.val)) convert hc using 1 ext x constructor · intro hx let t := Caratheodory.minCardFinsetOfMemConvexHull hx have ht : (t : Set E) ⊆ s := Caratheodory.minCardFinsetOfMemConvexHull_subseteq hx have hm : x ∈ convexHull ℝ (t : Set E) := Caratheodory.mem_minCardFinsetOfMemConvexHull hx have hi : AffineIndependent ℝ ((↑) : t → E) := Caratheodory.affineIndependent_minCardFinsetOfMemConvexHull hx have hn : Fintype.card t ≤ Module.finrank ℝ E + 1 := hi.card_le_finrank_succ.trans (Nat.add_le_add_right (Submodule.finrank_le _) 1) let k : Fin (Module.finrank ℝ E + 2) := ⟨Fintype.card t, by omega⟩ let e := (Fintype.equivFin t).symm obtain ⟨w,hw,hws,hxw⟩ := Finset.mem_convexHull'.mp hm let z : Fin k.val → s := fun i => ⟨(e i).val, ht (e i).property⟩ have hsum : ∑ i : Fin k.val, w (e i).val = 1 := by rw [show (∑ i : Fin k.val, w (e i).val) = ∑ j : t, w j.val from e.sum_comp (fun j : t => w j.val)] simpa only [Finset.sum_coe_sort] using hws let v : C k.val := (⟨Finsupp.equivFunOnFinite.symm (fun i => w (e i).val), fun i => by simpa using hw _ (e i).property, by simpa [Finsupp.sum_fintype] using hsum⟩, z) apply Set.mem_iUnion.mpr refine ⟨k,v,?_⟩ simp only [f, v, z, Finsupp.coe_equivFunOnFinite_symm] rw [show (∑ i : Fin k.val, w (e i).val • (e i).val) = ∑ j : t, w j.val • j.val from e.sum_comp (fun j : t => w j.val • j.val)] exact (Finset.sum_coe_sort t (fun j : E => w j • j)).trans hxw · intro hx obtain ⟨k,v,rfl⟩ := Set.mem_iUnion.mp hx exact mem_convexHull_of_exists_fintype (fun i => v.1.weights i) (fun i => (v.2 i).val) v.1.weights_nonneg v.1.total_of_fintype (fun i => (v.2 i).property) rfl end SpinAngle.CompactMixture namespace SpinAngle.CompactTwirl variable {G : Type*} [contextInstance3168_21 : Group G] [contextInstance3168_31 : TopologicalSpace G] [contextInstance3168_52 : IsTopologicalGroup G] [contextInstance3169_2 : CompactSpace G] [contextInstance3169_19 : MeasurableSpace G] [contextInstance3169_39 : BorelSpace G] variable {E : Type*} [contextInstance3170_21 : NormedAddCommGroup E] [contextInstance3170_44 : InnerProductSpace ℂ E] [contextInstance3171_2 : FiniteDimensional ℂ E] lemma twirl_finite_mixture (ρ : Representation ℂ G E) (hρ : Continuous (continuousAction ρ)) (a : E →L[ℂ] E) : ∃ (ι : Type) (_ : Fintype ι) (w : ι → ℝ) (g : ι → G), (∀ i, 0 ≤ w i) ∧ (∑ i, w i = 1) ∧ twirl (continuousAction ρ) a = ∑ i, w i • conjugate (continuousAction ρ) a (g i) := by let f := conjugate (continuousAction ρ) a have hf : Continuous f := continuous_conjugate _ hρ a have hc : IsCompact (Set.range f) := isCompact_range hf have hm : twirl (continuousAction ρ) a ∈ convexHull ℝ (Set.range f) := (convex_convexHull ℝ _).integral_mem (CompactMixture.convexHull_compact hc).isClosed (Filter.Eventually.of_forall fun g => subset_convexHull ℝ _ (Set.mem_range_self g)) (integrable_conjugate _ hρ a) obtain ⟨ι,hι,w,z,hw,hs,hz,he⟩ := mem_convexHull_iff_exists_fintype.mp hm let := hι choose g hg using hz refine ⟨ι,hι,w,g,hw,hs,?_⟩ rw [← he] apply Finset.sum_congr rfl intro i _ rw [← hg i] end SpinAngle.CompactTwirl namespace SpinAngle.SignedCarrier open scoped Classical TensorProduct BigOperators InnerProductSpace ComplexOrder Matrix.Norms.L2Operator MatrixOrder open SpinAngle.Young SpinAngle.Specht SignedTensor variable (a b : YoungDiagram) (q : ℕ) (ha : a.colLen 0 ≤ q) (hb : b.colLen 0 ≤ q) lemma dimension_eq : dimension a b q ha hb = Module.finrank ℂ (PolynomialCarrier.Carrier a q ha) * Module.finrank ℂ (PolynomialCarrier.Carrier b q hb) := Module.finrank_tensorProduct lemma dimension_pos : 0 < dimension a b q ha hb := by have : Nontrivial (Carrier a b q ha hb) := IsSimpleModule.nontrivial (MonoidAlgebra ℂ (Group q)) (rep a b q ha hb).asModule exact Module.finrank_pos theorem dimension_le : dimension a b q ha hb ≤ (Fintype.card (Cells a) + Fintype.card (Cells b) + 1) ^ (q * (q-1)) := by rw [dimension_eq] let p := Fintype.card (Cells a) + Fintype.card (Cells b) + 1 have hA : Module.finrank ℂ (PolynomialCarrier.Carrier a q ha) ≤ p ^ (q*(q-1)/2) := (PolynomialCarrier.ordinary_finrank_le a q ha).trans (Nat.pow_le_pow_left (by dsimp [p]; omega) _) have hB : Module.finrank ℂ (PolynomialCarrier.Carrier b q hb) ≤ p ^ (q*(q-1)/2) := (PolynomialCarrier.ordinary_finrank_le b q hb).trans (Nat.pow_le_pow_left (by dsimp [p]; omega) _) apply (Nat.mul_le_mul hA hB).trans rw [← pow_add] have he : q*(q-1)/2 + q*(q-1)/2 = q*(q-1) := by have hh := PolyDimension.twice_card_lowerIndex q rw [PolyDimension.card_lowerIndex] at hh omega rw [he] end SpinAngle.SignedCarrier namespace SpinAngle.SignedTensor open scoped Matrix.Norms.L2Operator MatrixOrder ComplexOrder variable {q : ℕ} def conjugateDensity (g : Group q) (A : Mat q × Mat q) : Mat q × Mat q := (g.1.val * A.1 * star g.1.val, g.2.val * A.2 * star g.2.val) lemma conjugateDensity_isEvenDensity (g : Group q) (A : Mat q × Mat q) (hA : IsEvenDensity A) : IsEvenDensity (conjugateDensity g A) := by refine ⟨?_, ?_, ?_⟩ · exact Matrix.nonneg_iff_posSemidef.mp (star_right_conjugate_nonneg hA.1.nonneg g.1.val) · exact Matrix.nonneg_iff_posSemidef.mp (star_right_conjugate_nonneg hA.2.1.nonneg g.2.val) · change (g.1.val * A.1 * star g.1.val).trace + (g.2.val * A.2 * star g.2.val).trace = 1 rw [Matrix.trace_mul_cycle, g.1.property.1, one_mul, Matrix.trace_mul_cycle, g.2.property.1, one_mul] exact hA.2.2 lemma block_conjugateDensity (g : Group q) (A : Mat q × Mat q) : block (conjugateDensity g A) = (blockGroup g).val * block A * (blockGroup g⁻¹).val := by change block (((g.1.val,g.2.val) * A) * star (g.1.val,g.2.val)) = _ rw [map_mul, map_mul, block_star, map_inv] rfl variable {τ : Type*} [contextInstance3254_21 : Fintype τ] [contextInstance3254_33 : DecidableEq τ] lemma tensor_conjugateDensity (g : Group q) (A : Mat q × Mat q) : LinearMap.toContinuousLinearMap (TensorRep.action (ι := τ) (block (conjugateDensity g A))) = CompactTwirl.conjugate (CompactTwirl.continuousAction (rep (τ := τ))) (LinearMap.toContinuousLinearMap (TensorRep.action (block A))) g := by apply ContinuousLinearMap.ext intro v change TensorRep.action (block (conjugateDensity g A)) v = TensorRep.action (blockGroup g).val (TensorRep.action (block A) (TensorRep.action (blockGroup g⁻¹).val v)) rw [block_conjugateDensity, map_mul, map_mul] rfl end SpinAngle.SignedTensor namespace SpinAngle.SignedTensor open scoped Matrix.Norms.L2Operator MatrixOrder ComplexOrder variable {q : ℕ} instance groupMeasurableSpace : MeasurableSpace (Group q) := borel _ instance groupBorelSpace : BorelSpace (Group q) := ⟨rfl⟩ variable [contextInstance3273_9 : Nonempty (Fin q)] instance groupCompactSpace : CompactSpace (Group q) := by let := unitaryGroup_compactSpace (Fin q) infer_instance end SpinAngle.SignedTensor namespace SpinAngle.SignedCarrier open scoped Classical TensorProduct BigOperators InnerProductSpace ComplexOrder Matrix.Norms.L2Operator MatrixOrder open SpinAngle.Young SpinAngle.Specht SignedTensor CompactTwirl variable (a b : YoungDiagram) (q : ℕ) (ha : a.colLen 0 ≤ q) (hb : b.colLen 0 ≤ q) variable {τ : Type*} [contextInstance3283_21 : Fintype τ] [contextInstance3283_33 : DecidableEq τ] lemma projection_isStarProjection : IsStarProjection (typeProjection (τ := τ) a b q ha hb) := isStarProjection_starProjection variable [contextInstance3288_9 : Nonempty (Fin q)] lemma twirl_lower (A : Mat q × Mat q) (hA : IsEvenDensity A) : ((LinearMap.trace ℂ (Carrier a b q ha hb) (hermitianAction a b q ha hb A hA.1.isHermitian hA.2.1.isHermitian)).re / (dimension a b q ha hb : ℝ)) • typeProjection (τ := τ) a b q ha hb ≤ twirl (continuousAction (SignedTensor.rep (q := q) (τ := τ))) (LinearMap.toContinuousLinearMap (TensorRep.action (block A))) := by exact isotypic_twirl_lower (E := Carrier a b q ha hb) (F := Raw (q := q) (τ := τ)) (rep a b q ha hb) SignedTensor.rep (continuous_rep a b q ha hb) SignedTensor.continuous_rep SignedTensor.rep_unitary (LinearMap.toContinuousLinearMap (hermitianAction a b q ha hb A hA.1.isHermitian hA.2.1.isHermitian)) (LinearMap.toContinuousLinearMap (TensorRep.action (block A))) ((ContinuousLinearMap.nonneg_iff_isPositive).mpr (hermitianAction_positive a b q ha hb A hA.1 hA.2.1)) ((ContinuousLinearMap.nonneg_iff_isPositive).mpr (TensorRep.action_positive (block A) (block_posSemidef A hA.1 hA.2.1))) (fun L x => (intertwiner_hermitian a b q ha hb L A hA.1.isHermitian hA.2.1.isHermitian x).symm) end SpinAngle.SignedCarrier noncomputable section namespace SpinAngle.ProjectionOrder variable {A : Type*} [contextInstance3309_21 : CStarAlgebra A] [contextInstance3309_38 : PartialOrder A] [contextInstance3309_55 : StarOrderedRing A] lemma rescale_trace_lower (P B : A) (hP : 0 ≤ P) (F t d : ℝ) (hd : 0 < d) (ht : Real.exp (-F) ≤ t) (h : (t / d) • P ≤ B) : P ≤ (Real.exp F * d) • B := by have hbase : (Real.exp (-F) / d) • P ≤ B := (smul_le_smul_of_nonneg_right (div_le_div_of_nonneg_right ht hd.le) hP).trans h have hs := smul_le_smul_of_nonneg_left hbase (mul_pos (Real.exp_pos F) hd).le have he : (Real.exp F * d) * (Real.exp (-F) / d) = 1 := by rw [← mul_div_assoc, mul_right_comm, mul_div_cancel_right₀ _ hd.ne', ← Real.exp_add] simp simpa only [smul_smul, he, one_smul] using hs end SpinAngle.ProjectionOrder namespace SpinAngle.MatrixBridge open scoped Matrix.Norms.L2Operator MatrixOrder ComplexOrder variable {n : Type*} [contextInstance3325_21 : Fintype n] [contextInstance3325_33 : DecidableEq n] lemma inverse_mono (a b : EuclideanSpace ℂ n →L[ℂ] EuclideanSpace ℂ n) (h : a ≤ b) : (Matrix.toEuclideanCLM (n := n) (𝕜 := ℂ)).symm a ≤ (Matrix.toEuclideanCLM (n := n) (𝕜 := ℂ)).symm b := by apply sub_nonneg.mp rw [← map_sub] exact map_nonneg (Matrix.toEuclideanCLM (n := n) (𝕜 := ℂ)).symm (sub_nonneg.mpr h) end SpinAngle.MatrixBridge namespace SpinAngle.SignedCarrier open scoped Classical TensorProduct BigOperators InnerProductSpace ComplexOrder Matrix.Norms.L2Operator MatrixOrder open SpinAngle.Young SpinAngle.Specht SignedTensor CompactTwirl variable (a b : YoungDiagram) (q : ℕ) (ha : a.colLen 0 ≤ q) (hb : b.colLen 0 ≤ q) variable {τ : Type*} [contextInstance3337_21 : Fintype τ] [contextInstance3337_33 : DecidableEq τ] [contextInstance3337_49 : Nonempty (Fin q)] theorem exists_density_mixture : ∃ (ι : Type) (_ : Fintype ι) (w : ι → ℝ) (R : ι → Mat q × Mat q), (∀ i, 0 ≤ w i) ∧ (∑ i, w i = 1) ∧ (∀ i, IsEvenDensity (R i)) ∧ typeProjection (τ := τ) a b q ha hb ≤ (Real.exp (entropy a b q) * (dimension a b q ha hb : ℝ)) • ∑ i, w i • LinearMap.toContinuousLinearMap (TensorRep.action (block (R i))) := by obtain ⟨A,hA,htr⟩ := exists_density_trace_lower a b q ha hb obtain ⟨ι,hι,w,g,hw,hs,he⟩ := twirl_finite_mixture (SignedTensor.rep (q := q) (τ := τ)) SignedTensor.continuous_rep (LinearMap.toContinuousLinearMap (TensorRep.action (block A))) let := hι refine ⟨ι,hι,w,fun i => conjugateDensity (g i) A,hw,hs, fun i => conjugateDensity_isEvenDensity (g i) A hA,?_⟩ have hh := twirl_lower (τ := τ) a b q ha hb A hA have hbound := ProjectionOrder.rescale_trace_lower _ _ (projection_isStarProjection (τ := τ) a b q ha hb).nonneg (entropy a b q) _ (dimension a b q ha hb : ℝ) (by exact_mod_cast dimension_pos a b q ha hb) htr hh rw [he] at hbound simpa only [tensor_conjugateDensity] using hbound end SpinAngle.SignedCarrier namespace SpinAngle.MatrixBridge open scoped Matrix.Norms.L2Operator MatrixOrder ComplexOrder variable {n : Type*} [contextInstance3364_21 : Fintype n] [contextInstance3364_33 : DecidableEq n] local notation "Φ" => (Matrix.toEuclideanCLM (n := n) (𝕜 := ℂ)) lemma inverse_real_smul (r : ℝ) (a : EuclideanSpace ℂ n →L[ℂ] EuclideanSpace ℂ n) : (Φ).symm (r • a) = r • (Φ).symm a := by change (Φ).symm ((r : ℂ) • a) = (r : ℂ) • (Φ).symm a exact map_smul (Φ).symm _ _ lemma forward_real_smul (r : ℝ) (a : Matrix n n ℂ) : Φ (r • a) = r • Φ a := by change Φ ((r : ℂ) • a) = (r : ℂ) • Φ a exact map_smul Φ _ _ lemma forward_norm (a : Matrix n n ℂ) : ‖Φ a‖ = ‖a‖ := rfl lemma inverse_norm (a : EuclideanSpace ℂ n →L[ℂ] EuclideanSpace ℂ n) : ‖(Φ).symm a‖ = ‖a‖ := by rw [← forward_norm, (Φ).apply_symm_apply] lemma inverse_toContinuous (a : Matrix n n ℂ) : (Φ).symm (LinearMap.toContinuousLinearMap (Matrix.toEuclideanLin a)) = a := by exact (Φ).symm_apply_apply a end SpinAngle.MatrixBridge namespace SpinAngle.SignedCarrier open scoped Classical TensorProduct BigOperators InnerProductSpace ComplexOrder Matrix.Norms.L2Operator MatrixOrder open SpinAngle.Young SpinAngle.Specht SignedTensor variable (a b : YoungDiagram) (q : ℕ) (ha : a.colLen 0 ≤ q) (hb : b.colLen 0 ≤ q) variable {τ : Type*} [contextInstance3392_21 : Fintype τ] [contextInstance3392_33 : DecidableEq τ] def matrixProjection : Matrix (τ → Spin q) (τ → Spin q) ℂ := (Matrix.toEuclideanCLM (n := τ → Spin q) (𝕜 := ℂ)).symm (typeProjection a b q ha hb) lemma matrixProjection_isStarProjection : IsStarProjection (matrixProjection (τ := τ) a b q ha hb) := by have hp := projection_isStarProjection (τ := τ) a b q ha hb constructor · change matrixProjection a b q ha hb * matrixProjection a b q ha hb = _ unfold matrixProjection rw [← map_mul, hp.isIdempotentElem.eq] · change star (matrixProjection a b q ha hb) = _ unfold matrixProjection have hs := (Matrix.toEuclideanCLM (n := τ → Spin q) (𝕜 := ℂ)).symm.map_star' (typeProjection a b q ha hb) exact hs.symm.trans (congrArg (Matrix.toEuclideanCLM (n := τ → Spin q) (𝕜 := ℂ)).symm hp.isSelfAdjoint.star_eq) lemma inverse_action (A : Matrix (Spin q) (Spin q) ℂ) : (Matrix.toEuclideanCLM (n := τ → Spin q) (𝕜 := ℂ)).symm (LinearMap.toContinuousLinearMap (TensorRep.action A)) = TensorRep.tensor (fun _ : τ => A) := MatrixBridge.inverse_toContinuous _ lemma matrix_density_upper (A : Mat q × Mat q) (hA : IsEvenDensity A) : ‖TensorRep.tensor (fun _ : τ => block A) * matrixProjection a b q ha hb‖ ≤ Real.exp (-entropy a b q) := by have h := global_density_upper (τ := τ) a b q ha hb A hA rw [← MatrixBridge.inverse_norm] at h simpa only [map_mul, inverse_action, matrixProjection] using h lemma density_projection_commute (A : Mat q × Mat q) (hA : IsEvenDensity A) : Commute (LinearMap.toContinuousLinearMap (TensorRep.action (ι := τ) (block A))) (typeProjection a b q ha hb) := by change _ * _ = _ * _ apply ContinuousLinearMap.ext intro v exact (Isotypic.projection_commute_natural (E := Carrier a b q ha hb) (F := Raw (q := q) (τ := τ)) (rep a b q ha hb) SignedTensor.rep (hermitianAction a b q ha hb A hA.1.isHermitian hA.2.1.isHermitian) (TensorRep.action (block A)) (TensorRep.action_positive (block A) (block_posSemidef A hA.1 hA.2.1)).isSymmetric (fun L x => (intertwiner_hermitian a b q ha hb L A hA.1.isHermitian hA.2.1.isHermitian x).symm) v).symm lemma matrix_density_commute (A : Mat q × Mat q) (hA : IsEvenDensity A) : Commute (TensorRep.tensor (fun _ : τ => block A)) (matrixProjection a b q ha hb) := by have h := (density_projection_commute (τ := τ) a b q ha hb A hA).map (Matrix.toEuclideanCLM (n := τ → Spin q) (𝕜 := ℂ)).symm simpa only [inverse_action, matrixProjection] using h variable [contextInstance3441_9 : Nonempty (Fin q)] theorem exists_matrix_density_mixture : ∃ (ι : Type) (_ : Fintype ι) (w : ι → ℝ) (R : ι → Mat q × Mat q), (∀ i, 0 ≤ w i) ∧ (∑ i, w i = 1) ∧ (∀ i, IsEvenDensity (R i)) ∧ matrixProjection (τ := τ) a b q ha hb ≤ (Real.exp (entropy a b q) * (dimension a b q ha hb : ℝ)) • ∑ i, w i • TensorRep.tensor (fun _ : τ => block (R i)) := by obtain ⟨ι,hι,w,R,hw,hs,hR,hdom⟩ := exists_density_mixture (τ := τ) a b q ha hb let := hι refine ⟨ι,hι,w,R,hw,hs,hR,?_⟩ have h := MatrixBridge.inverse_mono _ _ hdom simpa only [MatrixBridge.inverse_real_smul, map_sum, inverse_action, matrixProjection] using h end SpinAngle.SignedCarrier namespace SpinAngle.SignedTensor open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder variable {q : ℕ} lemma block_add (A B : Mat q × Mat q) : block (A+B) = block A + block B := by ext i j rcases i with ⟨i,bi⟩; rcases j with ⟨j,bj⟩ cases bi <;> cases bj <;> simp [block, Matrix.blockDiagonal_apply] lemma block_real_smul (r : ℝ) (A : Mat q × Mat q) : block (r • A) = r • block A := by ext i j rcases i with ⟨i,bi⟩; rcases j with ⟨j,bj⟩ cases bi <;> cases bj <;> simp [block, Matrix.blockDiagonal_apply] lemma block_zero : block (0 : Mat q × Mat q) = 0 := by ext i j rcases i with ⟨i,bi⟩; rcases j with ⟨j,bj⟩ cases bi <;> cases bj <;> simp [block, Matrix.blockDiagonal_apply] lemma block_sum {ι : Type*} (s : Finset ι) (A : ι → Mat q × Mat q) : block (∑ i ∈ s, A i) = ∑ i ∈ s, block (A i) := by classical induction s using Finset.induction_on with | empty => simp [block_zero] | insert i s hi ih => simp only [Finset.sum_insert hi, block_add, ih] lemma block_trace (A : Mat q × Mat q) : (block A).trace = A.1.trace + A.2.trace := by change (Matrix.blockDiagonal (fun b : Bool => cond b A.2 A.1)).trace = _ rw [Matrix.trace_blockDiagonal] simp [add_comm] variable {J : Type*} [contextInstance3487_21 : Fintype J] def evenAverage (A : J → Mat q × Mat q) : Mat q × Mat q := (Fintype.card J : ℝ)⁻¹ • ∑ j, A j lemma evenAverage_isEvenDensity [Nonempty J] (A : J → Mat q × Mat q) (hA : ∀ j, IsEvenDensity (A j)) : IsEvenDensity (evenAverage A) := by have hpos : 0 < (Fintype.card J : ℝ) := by exact_mod_cast Fintype.card_pos have he (A : Mat q × Mat q) : A.1.trace + A.2.trace = (block A).trace := (block_trace A).symm refine ⟨?_,?_,?_⟩ · apply Matrix.nonneg_iff_posSemidef.mp simp only [evenAverage, Prod.smul_fst, Prod.fst_sum] exact smul_nonneg (inv_nonneg.mpr hpos.le) (Finset.sum_nonneg fun j _ => (hA j).1.nonneg) · apply Matrix.nonneg_iff_posSemidef.mp simp only [evenAverage, Prod.smul_snd, Prod.snd_sum] exact smul_nonneg (inv_nonneg.mpr hpos.le) (Finset.sum_nonneg fun j _ => (hA j).2.1.nonneg) · rw [he] unfold evenAverage rw [block_real_smul, block_sum, Matrix.trace_smul, Matrix.trace_sum] simp only [block_trace, (hA _).2.2, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_one] rw [RCLike.real_smul_eq_coe_smul (K := ℂ), smul_eq_mul] norm_cast exact inv_mul_cancel₀ (by exact_mod_cast hpos.ne') end SpinAngle.SignedTensor noncomputable section open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder namespace SpinAngle.MatrixAnalysis variable {n : Type*} [contextInstance3521_21 : Fintype n] [contextInstance3521_33 : DecidableEq n] abbrev Mat (n : Type*) := Matrix n n ℂ def root (a : Mat n) : Mat n := cfc Real.sqrt a def invRoot (a : Mat n) : Mat n := cfc (fun x : ℝ => (Real.sqrt x)⁻¹) a def support (a : Mat n) : Mat n := cfc (fun x : ℝ => if x = 0 then 0 else 1) a lemma root_selfAdjoint (a : Mat n) : IsSelfAdjoint (root a) := cfc_predicate _ _ lemma root_nonneg (a : Mat n) : 0 ≤ root a := cfc_nonneg fun _ _ => Real.sqrt_nonneg _ lemma invRoot_selfAdjoint (a : Mat n) : IsSelfAdjoint (invRoot a) := cfc_predicate _ _ lemma invRoot_nonneg (a : Mat n) : 0 ≤ invRoot a := cfc_nonneg fun _ _ => inv_nonneg.mpr (Real.sqrt_nonneg _) lemma support_selfAdjoint (a : Mat n) : IsSelfAdjoint (support a) := cfc_predicate _ _ lemma root_mul_self (a : Mat n) (ha : 0 ≤ a) : root a * root a = a := by rw [root, ← cfc_mul _ _ _ (Matrix.finite_real_spectrum (A := a) |>.continuousOn _) (Matrix.finite_real_spectrum (A := a) |>.continuousOn _)] calc _ = cfc (fun x : ℝ => x) a := by apply cfc_congr intro x hx exact Real.mul_self_sqrt (spectrum_nonneg_of_nonneg ha hx) _ = a := cfc_id' ℝ (ha := .of_nonneg ha) lemma invRoot_mul_root (a : Mat n) (ha : 0 ≤ a) : invRoot a * root a = support a := by rw [invRoot, root, ← cfc_mul _ _ _ (Matrix.finite_real_spectrum (A := a) |>.continuousOn _) (Matrix.finite_real_spectrum (A := a) |>.continuousOn _)] apply cfc_congr intro x hx by_cases hx0 : x = 0 · simp [hx0] · have hs : Real.sqrt x ≠ 0 := ne_of_gt (Real.sqrt_pos.mpr (lt_of_le_of_ne (spectrum_nonneg_of_nonneg ha hx) (Ne.symm hx0))) simp [hx0, hs] lemma support_isStarProjection (a : Mat n) : IsStarProjection (support a) := by refine ⟨?_, support_selfAdjoint a⟩ change support a * support a = support a rw [support, ← cfc_mul _ _ _ (Matrix.finite_real_spectrum (A := a) |>.continuousOn _) (Matrix.finite_real_spectrum (A := a) |>.continuousOn _)] apply cfc_congr intro x _ dsimp only split_ifs <;> norm_num lemma mul_support (a : Mat n) (ha : 0 ≤ a) : a * support a = a := by conv_lhs => lhs; rw [← cfc_id' ℝ (a := a)] rw [support, ← cfc_mul _ _ _ (Matrix.finite_real_spectrum (A := a) |>.continuousOn _) (Matrix.finite_real_spectrum (A := a) |>.continuousOn _)] calc _ = cfc (fun x : ℝ => x) a := by apply cfc_congr intro x _ by_cases hx : x = 0 <;> simp [hx] _ = a := cfc_id' ℝ (ha := .of_nonneg ha) lemma root_mul_support (a : Mat n) : root a * support a = root a := by rw [root, support, ← cfc_mul _ _ _ (Matrix.finite_real_spectrum (A := a) |>.continuousOn _) (Matrix.finite_real_spectrum (A := a) |>.continuousOn _)] apply cfc_congr intro x _ by_cases hx : x = 0 <;> simp [hx] lemma root_commute {a b : Mat n} (h : Commute a b) : Commute (root a) b := h.cfc_real _ lemma invRoot_commute {a b : Mat n} (h : Commute a b) : Commute (invRoot a) b := h.cfc_real _ lemma support_commute {a b : Mat n} (h : Commute a b) : Commute (support a) b := h.cfc_real _ lemma norm_le_re_trace (a : Mat n) (ha : 0 ≤ a) : ‖a‖ ≤ (Matrix.trace a).re := by have hA := Matrix.nonneg_iff_posSemidef.mp ha rw [hA.isHermitian.trace_eq_sum_eigenvalues, Complex.re_sum] conv_lhs => rw [hA.isHermitian.spectral_theorem] simp only [Unitary.conjStarAlgAut_apply, ← Unitary.coe_star, CStarRing.norm_mul_coe_unitary, CStarRing.norm_coe_unitary_mul, Matrix.l2_opNorm_diagonal] apply (pi_norm_le_iff_of_nonneg (by exact Finset.sum_nonneg fun i _ => hA.eigenvalues_nonneg i)).mpr intro i simp only [Function.comp_apply, RCLike.norm_ofReal, abs_of_nonneg (hA.eigenvalues_nonneg i)] exact Finset.single_le_sum (fun j _ => hA.eigenvalues_nonneg j) (Finset.mem_univ i) lemma norm_sq_le_re_trace_star_mul_self (a : Mat n) : ‖a‖ ^ 2 ≤ (Matrix.trace (star a * a)).re := by rw [sq, ← CStarRing.norm_star_mul_self] exact norm_le_re_trace _ (star_mul_self_nonneg _) lemma root_mul_support_of_le_smul (a b : Mat n) (ha : 0 ≤ a) (hb : 0 ≤ b) (c : ℝ) (hba : b ≤ c • a) : root b * support a = root b := by let e := 1 - support a have he : IsStarProjection e := (support_isStarProjection a).one_sub have hae : a * e = 0 := by dsimp [e] rw [mul_sub, mul_one, mul_support a ha, sub_self] have hupper : star e * b * e ≤ 0 := by have hc := star_left_conjugate_le_conjugate hba e simpa only [mul_smul_comm, smul_mul_assoc, mul_assoc, hae, mul_zero, smul_zero] using hc have hzero : star e * b * e = 0 := le_antisymm hupper (star_left_conjugate_nonneg hb e) have hrootzero : star (root b * e) * (root b * e) = 0 := by rw [star_mul, (root_selfAdjoint b).star_eq] calc (star e * root b) * (root b * e) = star e * (root b * root b) * e := by simp only [mul_assoc] _ = 0 := by rw [root_mul_self b hb, hzero] have h := (CStarRing.star_mul_self_eq_zero_iff _).mp hrootzero dsimp [e] at h rw [mul_sub, mul_one] at h exact (sub_eq_zero.mp h).symm lemma invRoot_mul_self_mul_invRoot (a : Mat n) (ha : 0 ≤ a) : invRoot a * a * invRoot a = support a := by have hc : Commute (invRoot a) a := invRoot_commute (Commute.refl a) have hr : Commute (root a) (invRoot a) := root_commute hc.symm calc invRoot a * a * invRoot a = invRoot a * (root a * root a) * invRoot a := congrArg (fun x => invRoot a * x * invRoot a) (root_mul_self a ha).symm _ = (invRoot a * root a) * (root a * invRoot a) := by simp only [mul_assoc] _ = support a * support a := by rw [hr.eq, invRoot_mul_root a ha] _ = support a := (support_isStarProjection a).isIdempotentElem.eq lemma cell_trace_identity (a b c : Mat n) (ha : 0 ≤ a) (hc : 0 ≤ c) : (star (root a * invRoot b * root c) * (root a * invRoot b * root c)).trace = (invRoot b * a * invRoot b * c).trace := by have he : star (root a * invRoot b * root c) * (root a * invRoot b * root c) = root c * (invRoot b * a * invRoot b) * root c := by simp only [star_mul, (root_selfAdjoint a).star_eq, (root_selfAdjoint c).star_eq, (invRoot_selfAdjoint b).star_eq] calc root c * (invRoot b * root a) * (root a * invRoot b * root c) = root c * (invRoot b * (root a * root a) * invRoot b) * root c := by simp only [mul_assoc] _ = _ := by rw [root_mul_self a ha] rw [he, Matrix.trace_mul_comm (root c * (invRoot b * a * invRoot b)) (root c)] rw [← mul_assoc, root_mul_self c hc, Matrix.trace_mul_comm c] lemma cell_trace_nonneg (a b c : Mat n) (ha : 0 ≤ a) (hc : 0 ≤ c) : 0 ≤ (invRoot b * a * invRoot b * c).trace := by rw [← cell_trace_identity a b c ha hc] exact (Matrix.nonneg_iff_posSemidef.mp (star_mul_self_nonneg _)).trace_nonneg lemma re_trace_nonneg {n : Type*} [contextInstance3521_21 : Fintype n] [DecidableEq n] (a : Mat n) (ha : 0 ≤ a) : 0 ≤ a.trace.re := (Complex.nonneg_iff.mp ((Matrix.nonneg_iff_posSemidef.mp ha).trace_nonneg)).1 lemma re_trace_mono {a b : Mat n} (hab : a ≤ b) : a.trace.re ≤ b.trace.re := by have h := re_trace_nonneg (b - a) (sub_nonneg.mpr hab) simpa only [Matrix.trace_sub, Complex.sub_re, sub_nonneg] using h lemma re_trace_projection_mul_le (q b : Mat n) (hq : IsStarProjection q) (hb : 0 ≤ b) : (q * b).trace.re ≤ b.trace.re := by have h := re_trace_mono (star_left_conjugate_le_conjugate hq.le_one (root b)) rw [mul_one, (root_selfAdjoint b).star_eq, root_mul_self b hb] at h rw [Matrix.trace_mul_comm (root b * q) (root b), ← mul_assoc, root_mul_self b hb, Matrix.trace_mul_comm b q] at h exact h lemma cell_norm_sq_le_trace (a b c : Mat n) (ha : 0 ≤ a) (hc : 0 ≤ c) : ‖root a * invRoot b * root c‖ ^ 2 ≤ (invRoot b * a * invRoot b * c).trace.re := by have h := norm_sq_le_re_trace_star_mul_self (root a * invRoot b * root c) rw [cell_trace_identity a b c ha hc] at h exact h end SpinAngle.MatrixAnalysis namespace SpinAngle.TensorMatrix open SpinAngle.MatrixAnalysis variable {ι n : Type*} [contextInstance3707_23 : Fintype ι] [contextInstance3707_35 : DecidableEq ι] [contextInstance3707_51 : Fintype n] [contextInstance3707_63 : DecidableEq n] def tensor (a : ι → Mat n) : Mat (ι → n) := fun x y => ∏ i, a i (x i) (y i) lemma tensor_star {ι : Type*} {n : Type*} [contextInstance3707_23 : Fintype ι] [DecidableEq ι] [Fintype n] [DecidableEq n] (a : ι → Mat n) : tensor (fun i => star (a i)) = star (tensor a) := by ext x y simp [tensor, Matrix.star_apply] lemma tensor_mul {ι : Type*} {n : Type*} [contextInstance3707_23 : Fintype ι] [contextInstance3707_35 : DecidableEq ι] [contextInstance3707_51 : Fintype n] [DecidableEq n] (a b : ι → Mat n) : tensor (fun i => a i * b i) = tensor a * tensor b := by ext x y simp only [tensor, Matrix.mul_apply, ← Finset.prod_mul_distrib] symm simpa using (Finset.sum_prod_piFinset (ι := ι) Finset.univ (fun i j => a i (x i) j * b i j (y i))) lemma tensor_one {ι : Type*} {n : Type*} [contextInstance3707_23 : Fintype ι] [DecidableEq ι] [Fintype n] [contextInstance3707_63 : DecidableEq n] : tensor (fun _ : ι => (1 : Mat n)) = 1 := by ext x y classical by_cases h : x = y · subst y simp [tensor] · have hxy : ∃ i, x i ≠ y i := Function.ne_iff.mp h obtain ⟨i, hi⟩ := hxy simp only [tensor, Matrix.one_apply, ite_eq_right h] apply Finset.prod_eq_zero (Finset.mem_univ i) simp [hi] lemma tensor_trace {ι : Type*} {n : Type*} [contextInstance3707_23 : Fintype ι] [contextInstance3707_35 : DecidableEq ι] [contextInstance3707_51 : Fintype n] [DecidableEq n] (a : ι → Mat n) : (tensor a).trace = ∏ i, (a i).trace := by simp only [Matrix.trace, Matrix.diag, tensor] simpa using (Finset.sum_prod_piFinset (ι := ι) Finset.univ (fun i j => a i j j)) lemma tensor_nonneg (a : ι → Mat n) (ha : ∀ i, 0 ≤ a i) : 0 ≤ tensor a := by have hr : tensor a = star (tensor (fun i => root (a i))) * tensor (fun i => root (a i)) := by rw [← tensor_star, ← tensor_mul] congr 1 funext i rw [(root_selfAdjoint (a i)).star_eq, root_mul_self _ (ha i)] rw [hr] exact star_mul_self_nonneg _ lemma tensor_selfAdjoint (a : ι → Mat n) (ha : ∀ i, IsSelfAdjoint (a i)) : IsSelfAdjoint (tensor a) := by change star (tensor a) = tensor a rw [← tensor_star] congr 1 funext i exact (ha i).star_eq lemma tensor_norm_sq_le_trace_product (a : ι → Mat n) : ‖tensor a‖ ^ 2 ≤ (∏ i, (star (a i) * a i).trace).re := by have h := norm_sq_le_re_trace_star_mul_self (tensor a) rw [← tensor_star, ← tensor_mul, tensor_trace] at h exact h lemma tensor_norm_sq_le_re_trace_product (a : ι → Mat n) : ‖tensor a‖ ^ 2 ≤ ∏ i, (star (a i) * a i).trace.re := by have hreal (i : ι) : (star (a i) * a i).trace = ((star (a i) * a i).trace.re : ℂ) := Complex.eq_re_of_ofReal_le (r := 0) (by exact (Matrix.nonneg_iff_posSemidef.mp (star_mul_self_nonneg _)).trace_nonneg) calc _ ≤ (∏ i, (star (a i) * a i).trace).re := tensor_norm_sq_le_trace_product a _ = (∏ i, ((star (a i) * a i).trace.re : ℂ)).re := congrArg Complex.re (Finset.prod_congr rfl fun i _ => hreal i) _ = _ := by rw [← Complex.ofReal_prod, Complex.ofReal_re] lemma tensor_cell_norm_sq_le_product (a b c : ι → Mat n) (ha : ∀ i, 0 ≤ a i) (hc : ∀ i, 0 ≤ c i) : ‖tensor (fun i => root (a i) * invRoot (b i) * root (c i))‖ ^ 2 ≤ ∏ i, (invRoot (b i) * a i * invRoot (b i) * c i).trace.re := by have h := tensor_norm_sq_le_re_trace_product (fun i => root (a i) * invRoot (b i) * root (c i)) simpa only [cell_trace_identity _ _ _ (ha _) (hc _)] using h end SpinAngle.TensorMatrix noncomputable section open scoped BigOperators namespace SpinAngle def missingCellFactor (m b : ℕ) : ℝ := if b < m then ((m : ℝ) / (m - b : ℕ)) ^ (m - b) else 1 def missingCellProductFactor {ι : Type*} [Fintype ι] (m : ℕ) (b : ι → ℕ) : ℝ := ∏ i, missingCellFactor m (b i) lemma missingCellFactor_nonneg (m b : ℕ) : 0 ≤ missingCellFactor m b := by unfold missingCellFactor split_ifs <;> positivity lemma missingCellFactor_le_exp (m b : ℕ) (hb : b ≤ m) : missingCellFactor m b ≤ Real.exp b := by unfold missingCellFactor split_ifs with h · have hk : 0 < m - b := Nat.sub_pos_of_lt h have hkR : (0 : ℝ) < (m - b : ℕ) := by exact_mod_cast hk have hmR : 0 < (m : ℝ) := by exact_mod_cast (lt_of_le_of_lt (Nat.zero_le b) h) have hratio : 0 < (m : ℝ) / (m - b : ℕ) := div_pos hmR hkR apply (Real.log_le_iff_le_exp (pow_pos hratio _)).mp rw [Real.log_pow] calc (m - b : ℕ) * Real.log ((m : ℝ) / (m - b : ℕ)) ≤ (m - b : ℕ) * ((m : ℝ) / (m - b : ℕ) - 1) := mul_le_mul_of_nonneg_left (Real.log_le_sub_one_of_pos hratio) hkR.le _ = (b : ℝ) := by rw [mul_sub, mul_div_cancel₀ _ hkR.ne', mul_one, Nat.cast_sub hb] ring · exact Real.one_le_exp (Nat.cast_nonneg b) theorem missingCellProduct {ι : Type*} [Fintype ι] (m : ℕ) (b : ι → ℕ) (hb : ∀ i, b i ≤ m) : missingCellProductFactor m b ≤ Real.exp (∑ i, (b i : ℝ)) := by unfold missingCellProductFactor calc ∏ i, missingCellFactor m (b i) ≤ ∏ i, Real.exp (b i : ℝ) := Finset.prod_le_prod₀ (fun i _ => missingCellFactor_nonneg m (b i)) (fun i _ => missingCellFactor_le_exp m (b i) (hb i)) _ = Real.exp (∑ i, (b i : ℝ)) := (Real.exp_sum _ _).symm end SpinAngle namespace SpinAngle open scoped BigOperators variable {ι : Type*} [contextInstance3841_21 : Fintype ι] [contextInstance3841_33 : DecidableEq ι] lemma retained_product_le {ι : Type*} [contextInstance3841_21 : Fintype ι] [DecidableEq ι] (t : Finset ι) (z : ι → ℝ) (hz : ∀ i, 0 ≤ z i) (hs : ∑ i, z i ≤ Fintype.card ι) : ∏ i ∈ t, z i ≤ ((Fintype.card ι : ℝ) / t.card) ^ t.card := by by_cases ht : t.Nonempty · have hkR : 0 < (t.card : ℝ) := by exact_mod_cast ht.card_pos have hmR : 0 < (Fintype.card ι : ℝ) := hkR.trans_le (by exact_mod_cast Finset.card_le_univ t) let c : ℝ := Fintype.card ι / t.card have hc : 0 < c := div_pos hmR hkR have hsum : ∑ i ∈ t, z i ≤ Fintype.card ι := (Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (by intro i _ _; exact hz i)).trans hs have hnorm : ∑ i ∈ t, z i / c ≤ t.card := by rw [← Finset.sum_div] apply (div_le_iff₀ hc).mpr dsimp [c] rwa [mul_div_cancel₀ _ hkR.ne'] have hn : ∏ i ∈ t, z i / c ≤ 1 := calc _ ≤ ∏ i ∈ t, Real.exp (z i / c - 1) := Finset.prod_le_prod₀ (fun i _ => div_nonneg (hz i) hc.le) fun i _ => by have h := Real.add_one_le_exp (z i / c - 1) linarith _ = Real.exp (∑ i ∈ t, (z i / c - 1)) := (Real.exp_sum _ _).symm _ ≤ Real.exp 0 := by apply Real.exp_le_exp.mpr rw [Finset.sum_sub_distrib] simpa using sub_nonpos.mpr hnorm _ = 1 := Real.exp_zero rw [Finset.prod_div_distrib, Finset.prod_const] at hn exact (div_le_one (pow_pos hc _)).mp hn · have he : t = ∅ := Finset.not_nonempty_iff_eq_empty.mp ht simp [he] variable {R C : Type*} [contextInstance3877_23 : Fintype R] [contextInstance3877_35 : DecidableEq R] [contextInstance3877_51 : Fintype C] [contextInstance3877_63 : DecidableEq C] def occupiedRow (W : Finset (R × C)) (i : R) : Finset C := Finset.univ.filter (fun j => (i, j) ∈ W) def missingInRow (W : Finset (R × C)) (i : R) : ℕ := Fintype.card C - (occupiedRow W i).card lemma occupiedRow_card_le {R : Type*} {C : Type*} [Fintype R] [contextInstance3877_35 : DecidableEq R] [contextInstance3877_51 : Fintype C] [contextInstance3877_63 : DecidableEq C] (W : Finset (R × C)) (i : R) : (occupiedRow W i).card ≤ Fintype.card C := Finset.card_le_univ _ lemma missingInRow_le {R : Type*} {C : Type*} [Fintype R] [contextInstance3877_35 : DecidableEq R] [contextInstance3877_51 : Fintype C] [contextInstance3877_63 : DecidableEq C] (W : Finset (R × C)) (i : R) : missingInRow W i ≤ Fintype.card C := Nat.sub_le _ _ lemma retained_factor_eq_missingCellFactor (W : Finset (R × C)) (i : R) : ((Fintype.card C : ℝ) / (occupiedRow W i).card) ^ (occupiedRow W i).card = missingCellFactor (Fintype.card C) (missingInRow W i) := by have hc := occupiedRow_card_le W i unfold missingInRow missingCellFactor rw [Nat.sub_sub_self hc] split_ifs with h · rfl · have hk : (occupiedRow W i).card = 0 := by omega simp [hk] lemma prod_occupiedRow {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} {C : Type*} [contextInstance3877_23 : Fintype R] [contextInstance3877_35 : DecidableEq R] [contextInstance3877_51 : Fintype C] [contextInstance3877_63 : DecidableEq C] (W : Finset (R × C)) (z : R → C → ℝ) : (∏ i, ∏ j ∈ occupiedRow W i, z i j) = ∏ x ∈ W, z x.1 x.2 := by simp only [occupiedRow, Finset.prod_filter] rw [← Finset.prod_product (f := fun x : R × C => if x ∈ W then z x.1 x.2 else 1)] simp only [Finset.univ_product_univ, ← Finset.prod_filter] congr 1 ext x simp lemma board_product_le (W : Finset (R × C)) (z : R → C → ℝ) (hz : ∀ i j, 0 ≤ z i j) (hs : ∀ i, ∑ j, z i j ≤ Fintype.card C) : (∏ x ∈ W, z x.1 x.2) ≤ missingCellProductFactor (Fintype.card C) (missingInRow W) := by rw [← prod_occupiedRow (ι := R) W z] unfold missingCellProductFactor apply Finset.prod_le_prod₀ · intro i _ exact Finset.prod_nonneg fun j _ => hz i j · intro i _ rw [← retained_factor_eq_missingCellFactor] exact retained_product_le (occupiedRow W i) (z i) (hz i) (hs i) end SpinAngle namespace SpinAngle.MatrixAnalysis variable {n C : Type*} [contextInstance3931_23 : Fintype n] [contextInstance3931_35 : DecidableEq n] [contextInstance3931_51 : Fintype C] [contextInstance3931_63 : DecidableEq C] structure Density (n : Type*) [Fintype n] [DecidableEq n] where matrix : Mat n nonneg : 0 ≤ matrix trace_eq_one : matrix.trace = 1 instance : Coe (Density n) (Mat n) := ⟨Density.matrix⟩ def average (r : C → Density n) : Mat n := (Fintype.card C : ℝ)⁻¹ • ∑ j, (r j).matrix lemma average_nonneg {n : Type*} {C : Type*} [contextInstance3931_23 : Fintype n] [contextInstance3931_35 : DecidableEq n] [contextInstance3931_51 : Fintype C] [DecidableEq C] (r : C → Density n) : 0 ≤ average r := smul_nonneg (inv_nonneg.mpr (Nat.cast_nonneg _)) (Finset.sum_nonneg fun j _ => (r j).nonneg) lemma sum_eq_smul_average {n : Type*} {C : Type*} [contextInstance3931_23 : Fintype n] [contextInstance3931_35 : DecidableEq n] [contextInstance3931_51 : Fintype C] [DecidableEq C] [Nonempty C] (r : C → Density n) : (∑ j, (r j).matrix) = (Fintype.card C : ℝ) • average r := by unfold average rw [smul_smul, mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero (α := C)), one_smul] lemma le_smul_average [Nonempty C] (r : C → Density n) (j : C) : (r j).matrix ≤ (Fintype.card C : ℝ) • average r := by rw [← sum_eq_smul_average] exact Finset.single_le_sum (fun k _ => (r k).nonneg) (Finset.mem_univ j) lemma support_cell_trace_sum [Nonempty C] (r : C → Density n) (s : Density n) : ∑ j, (invRoot (average r) * (r j).matrix * invRoot (average r) * s.matrix).trace.re ≤ Fintype.card C := by have he : (∑ j, invRoot (average r) * (r j).matrix * invRoot (average r) * s.matrix) = (Fintype.card C : ℝ) • (support (average r) * s.matrix) := by rw [← Finset.sum_mul, ← Finset.sum_mul, ← Finset.mul_sum, sum_eq_smul_average r, mul_smul_comm, smul_mul_assoc, smul_mul_assoc, invRoot_mul_self_mul_invRoot _ (average_nonneg r)] have ht := congrArg (fun a : Mat n => a.trace.re) he simp only [Matrix.trace_sum, Complex.re_sum, Matrix.trace_smul, Complex.smul_re, smul_eq_mul] at ht rw [ht] have h := re_trace_projection_mul_le (support (average r)) s.matrix (support_isStarProjection _) s.nonneg rw [s.trace_eq_one, Complex.one_re] at h simpa using mul_le_mul_of_nonneg_left h (Nat.cast_nonneg (Fintype.card C)) end SpinAngle.MatrixAnalysis namespace SpinAngle open MatrixAnalysis TensorMatrix variable {n R C : Type*} [contextInstance3979_25 : Fintype n] [contextInstance3979_37 : DecidableEq n] [contextInstance3980_2 : Fintype R] [contextInstance3980_14 : DecidableEq R] [contextInstance3980_30 : Fintype C] [contextInstance3980_42 : DecidableEq C] [contextInstance3980_58 : Nonempty C] theorem normalized_board_bound (W : Finset (R × C)) (r : C → Density n) (s : R → Density n) : ‖tensor (fun w : W => root (r w.val.2).matrix * invRoot (average r) * root (s w.val.1).matrix)‖ ^ 2 ≤ missingCellProductFactor (Fintype.card C) (missingInRow W) := by let z : R → C → ℝ := fun i j => (invRoot (average r) * (r j).matrix * invRoot (average r) * (s i).matrix).trace.re have hz : ∀ i j, 0 ≤ z i j := fun i j => (Complex.nonneg_iff.mp (cell_trace_nonneg _ _ _ (r j).nonneg (s i).nonneg)).1 have hs : ∀ i, ∑ j, z i j ≤ Fintype.card C := fun i => support_cell_trace_sum r (s i) calc _ ≤ ∏ w : W, z w.val.1 w.val.2 := tensor_cell_norm_sq_le_product _ _ _ (fun w => (r w.val.2).nonneg) (fun w => (s w.val.1).nonneg) _ = ∏ w ∈ W, z w.1 w.2 := (Finset.prod_subtype W (by intro w; rfl) (fun w => z w.1 w.2)).symm _ ≤ _ := board_product_le W z hz hs end SpinAngle noncomputable section open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder namespace SpinAngle.DependentTensor variable {ι : Type*} [contextInstance4012_21 : Fintype ι] [contextInstance4012_33 : DecidableEq ι] variable {ν : ι → Type*} [contextInstance4013_25 : ∀ i, Fintype (ν i)] [contextInstance4013_46 : ∀ i, DecidableEq (ν i)] abbrev Mat (n : Type*) := Matrix n n ℂ def tensor (A : ∀ i, Mat (ν i)) : Mat (∀ i, ν i) := fun x y => ∏ i, A i (x i) (y i) lemma tensor_star {ι : Type*} [contextInstance4012_21 : Fintype ι] [DecidableEq ι] {ν : ι → Type*} [∀ i, Fintype (ν i)] [∀ i, DecidableEq (ν i)] (A : ∀ i, Mat (ν i)) : tensor (fun i => star (A i)) = star (tensor A) := by ext x y simp [tensor, Matrix.star_apply] lemma tensor_mul {ι : Type*} [contextInstance4012_21 : Fintype ι] [contextInstance4012_33 : DecidableEq ι] {ν : ι → Type*} [contextInstance4013_25 : ∀ i, Fintype (ν i)] [∀ i, DecidableEq (ν i)] (A B : ∀ i, Mat (ν i)) : tensor (fun i => A i * B i) = tensor A * tensor B := by ext x y simp only [tensor, Matrix.mul_apply, ← Finset.prod_mul_distrib] exact (Fintype.prod_sum (fun i (j : ν i) => A i (x i) j * B i j (y i))) lemma tensor_one {ι : Type*} [contextInstance4012_21 : Fintype ι] [DecidableEq ι] {ν : ι → Type*} [∀ i, Fintype (ν i)] [contextInstance4013_46 : ∀ i, DecidableEq (ν i)] : tensor (fun i : ι => (1 : Mat (ν i))) = 1 := by ext x y classical by_cases h : x = y · subst y; simp [tensor] · obtain ⟨i,hi⟩ := Function.ne_iff.mp h simp only [tensor, Matrix.one_apply, ite_eq_right h] apply Finset.prod_eq_zero (Finset.mem_univ i) simp [hi] lemma tensor_nonneg (A : ∀ i, Mat (ν i)) (hA : ∀ i, 0 ≤ A i) : 0 ≤ tensor A := by have he : tensor A = star (tensor (fun i => CFC.sqrt (A i))) * tensor (fun i => CFC.sqrt (A i)) := by rw [← tensor_star, ← tensor_mul] congr 1 funext i rw [(IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg (A i))).star_eq, CFC.sqrt_mul_sqrt_self (A i) (hA i)] rw [he] exact star_mul_self_nonneg _ lemma tensor_projection (P : ∀ i, Mat (ν i)) (hP : ∀ i, IsStarProjection (P i)) : IsStarProjection (tensor P) := by constructor · change tensor P * tensor P = tensor P rw [← tensor_mul] congr 1 funext i exact (hP i).isIdempotentElem.eq · change star (tensor P) = tensor P rw [← tensor_star] congr 1 funext i exact (hP i).isSelfAdjoint.star_eq lemma tensor_sum {ι : Type*} [contextInstance4012_21 : Fintype ι] [contextInstance4012_33 : DecidableEq ι] {ν : ι → Type*} [∀ i, Fintype (ν i)] [∀ i, DecidableEq (ν i)] {κ : ι → Type*} [∀ i, Fintype (κ i)] (A : ∀ i, κ i → Mat (ν i)) : tensor (fun i => ∑ k, A i k) = ∑ f : ∀ i, κ i, tensor (fun i => A i (f i)) := by ext x y simp only [tensor, Matrix.sum_apply] exact (Fintype.prod_sum (fun i (k : κ i) => A i k (x i) (y i))) lemma tensor_smul {ι : Type*} [contextInstance4012_21 : Fintype ι] [DecidableEq ι] {ν : ι → Type*} [∀ i, Fintype (ν i)] [∀ i, DecidableEq (ν i)] (w : ι → ℝ) (A : ∀ i, Mat (ν i)) : tensor (fun i => w i • A i) = (∏ i, w i) • tensor A := by ext x y simp only [tensor, Matrix.smul_apply, RCLike.real_smul_eq_coe_smul (K := ℂ), smul_eq_mul, Finset.prod_mul_distrib] change (∏ i, (w i : ℂ)) * _ = ((∏ i, w i : ℝ) : ℂ) * _ rw [Complex.ofReal_prod] lemma tensor_sum_smul {κ : ι → Type*} [∀ i, Fintype (κ i)] (w : ∀ i, κ i → ℝ) (A : ∀ i, κ i → Mat (ν i)) : tensor (fun i => ∑ k, w i k • A i k) = ∑ f : ∀ i, κ i, (∏ i, w i (f i)) • tensor (fun i => A i (f i)) := by rw [tensor_sum] simp only [tensor_smul] lemma tensor_mono (A B : ∀ i, Mat (ν i)) (hA : ∀ i, 0 ≤ A i) (hAB : ∀ i, A i ≤ B i) : tensor A ≤ tensor B := by let M (i : ι) (b : Bool) : Mat (ν i) := if b then B i - A i else A i have he (i : ι) : (∑ b : Bool, M i b) = B i := by simp [M] have hm (i : ι) (b : Bool) : 0 ≤ M i b := by cases b · exact hA i · exact sub_nonneg.mpr (hAB i) rw [show B = (fun i => ∑ b : Bool, M i b) by funext i; exact (he i).symm, tensor_sum] have h := Finset.single_le_sum (s := (Finset.univ : Finset (ι → Bool))) (f := fun f => tensor (fun i => M i (f i))) (fun f _ => tensor_nonneg _ (fun i => hm i (f i))) (Finset.mem_univ (fun _ => false)) exact h end SpinAngle.DependentTensor namespace SpinAngle.BlockTensor open DependentTensor variable {τ J n : Type*} [contextInstance4107_25 : Fintype τ] [contextInstance4107_37 : DecidableEq τ] [contextInstance4107_53 : Fintype J] [contextInstance4107_65 : DecidableEq J] [contextInstance4108_2 : Fintype n] [contextInstance4108_14 : DecidableEq n] variable (f : τ → J) abbrev Fiber (j : J) := {i : τ // f i = j} def splitEquiv : (τ → n) ≃ (∀ j : J, Fiber f j → n) where toFun x j i := x i.val invFun y i := y (f i) ⟨i,rfl⟩ left_inv x := rfl right_inv y := by funext j i rcases i with ⟨i,hi⟩ subst j rfl def tensor (A : ∀ j, Mat (Fiber f j → n)) : Mat (τ → n) := fun x y => DependentTensor.tensor A (splitEquiv f x) (splitEquiv f y) lemma tensor_mul (A B : ∀ j, Mat (Fiber f j → n)) : tensor f (fun j => A j * B j) = tensor f A * tensor f B := by change (DependentTensor.tensor (fun j => A j * B j)).submatrix (splitEquiv f) (splitEquiv f) = _ rw [DependentTensor.tensor_mul] exact (Matrix.submatrix_mul_equiv _ _ (splitEquiv f) (splitEquiv f) (splitEquiv f)).symm lemma tensor_star {τ : Type*} {J : Type*} {n : Type*} [Fintype τ] [DecidableEq τ] [contextInstance4107_53 : Fintype J] [DecidableEq J] [Fintype n] [DecidableEq n] (f : τ → J) (A : ∀ j, Mat (Fiber f j → n)) : tensor f (fun j => star (A j)) = star (tensor f A) := by ext x y simp [tensor, DependentTensor.tensor, Matrix.star_apply] lemma tensor_one : tensor f (fun j => (1 : Mat (Fiber f j → n))) = 1 := by ext x y change DependentTensor.tensor (fun j => (1 : Mat (Fiber f j → n))) _ _ = _ rw [DependentTensor.tensor_one] simp only [Matrix.one_apply, Equiv.apply_eq_iff_eq] lemma tensor_nonneg (A : ∀ j, Mat (Fiber f j → n)) (hA : ∀ j, 0 ≤ A j) : 0 ≤ tensor f A := by exact (Matrix.nonneg_iff_posSemidef.mp (DependentTensor.tensor_nonneg A hA)).submatrix (splitEquiv f) |>.nonneg lemma tensor_mono (A B : ∀ j, Mat (Fiber f j → n)) (hA : ∀ j, 0 ≤ A j) (hAB : ∀ j, A j ≤ B j) : tensor f A ≤ tensor f B := by have he := DependentTensor.tensor_mono A B hA hAB have hn := (Matrix.nonneg_iff_posSemidef.mp (sub_nonneg.mpr he)).submatrix (splitEquiv f) |>.nonneg exact sub_nonneg.mp hn lemma tensor_projection (P : ∀ j, Mat (Fiber f j → n)) (hP : ∀ j, IsStarProjection (P j)) : IsStarProjection (tensor f P) := by constructor · change tensor f P * tensor f P = tensor f P rw [← tensor_mul] congr 1 funext j exact (hP j).isIdempotentElem.eq · change star (tensor f P) = tensor f P rw [← tensor_star] congr 1 funext j exact (hP j).isSelfAdjoint.star_eq lemma tensor_sum_smul {κ : J → Type*} [∀ j, Fintype (κ j)] (w : ∀ j, κ j → ℝ) (A : ∀ j, κ j → Mat (Fiber f j → n)) : tensor f (fun j => ∑ k, w j k • A j k) = ∑ k : ∀ j, κ j, (∏ j, w j (k j)) • tensor f (fun j => A j (k j)) := by ext x y change DependentTensor.tensor (fun j => ∑ k, w j k • A j k) _ _ = _ rw [DependentTensor.tensor_sum_smul] simp only [Matrix.sum_apply, Matrix.smul_apply, tensor] end SpinAngle.BlockTensor namespace SpinAngle.BlockTensor open DependentTensor variable {τ J n : Type*} [contextInstance4180_25 : Fintype τ] [contextInstance4180_37 : DecidableEq τ] [contextInstance4180_53 : Fintype J] [contextInstance4180_65 : DecidableEq J] [contextInstance4181_2 : Fintype n] [contextInstance4181_14 : DecidableEq n] variable (f : τ → J) lemma prod_fibers {τ : Type*} {J : Type*} [contextInstance4180_25 : Fintype τ] [DecidableEq τ] [contextInstance4180_53 : Fintype J] [contextInstance4180_65 : DecidableEq J] (f : τ → J) {M : Type*} [CommMonoid M] (a : τ → M) : (∏ j : J, ∏ i : Fiber f j, a i.val) = ∏ i, a i := by calc _ = ∏ z : Sigma (Fiber f), a z.2.val := (Fintype.prod_sigma (fun z : Sigma (Fiber f) => a z.2.val)).symm _ = _ := (Equiv.sigmaFiberEquiv f).prod_comp a lemma tensor_cells {τ : Type*} {J : Type*} {n : Type*} [contextInstance4180_25 : Fintype τ] [contextInstance4180_37 : DecidableEq τ] [contextInstance4180_53 : Fintype J] [contextInstance4180_65 : DecidableEq J] [Fintype n] [DecidableEq n] (f : τ → J) (A : τ → Mat n) : tensor f (fun j => DependentTensor.tensor (fun i : Fiber f j => A i.val)) = DependentTensor.tensor A := by ext x y change (∏ j, ∏ i : Fiber f j, A i.val (x i.val) (y i.val)) = _ exact prod_fibers f (fun i => A i (x i) (y i)) lemma tensor_commute (A B : ∀ j, Mat (Fiber f j → n)) (h : ∀ j, Commute (A j) (B j)) : Commute (tensor f A) (tensor f B) := by change tensor f A * tensor f B = tensor f B * tensor f A rw [← tensor_mul, ← tensor_mul] congr 1 funext j exact (h j).eq end SpinAngle.BlockTensor namespace SpinAngle.TensorMatrix open MatrixAnalysis open scoped Matrix.Norms.L2Operator MatrixOrder ComplexOrder variable {ι n : Type*} [contextInstance4214_23 : Fintype ι] [contextInstance4214_35 : DecidableEq ι] [contextInstance4214_51 : Fintype n] [contextInstance4214_63 : DecidableEq n] lemma tensor_root_square (A : ι → Mat n) (hA : ∀ i, 0 ≤ A i) : tensor (fun i => root (A i)) * tensor (fun i => root (A i)) = tensor A := by rw [← tensor_mul] congr 1 funext i exact root_mul_self _ (hA i) lemma tensor_root (A : ι → Mat n) (hA : ∀ i, 0 ≤ A i) : tensor (fun i => root (A i)) = root (tensor A) := by apply (CFC.mul_self_eq_mul_self_iff _ _ (tensor_nonneg _ (fun i => root_nonneg (A i))) (root_nonneg _)).mp rw [tensor_root_square A hA, root_mul_self _ (tensor_nonneg A hA)] lemma tensor_invRoot_mul_root (A : ι → Mat n) (hA : ∀ i, 0 ≤ A i) : tensor (fun i => invRoot (A i)) * tensor (fun i => root (A i)) = tensor (fun i => support (A i)) := by rw [← tensor_mul] congr 1 funext i exact invRoot_mul_root _ (hA i) lemma tensor_root_mul_support_of_le_smul (A B : ι → Mat n) (hA : ∀ i, 0 ≤ A i) (hB : ∀ i, 0 ≤ B i) (c : ℝ) (hAB : ∀ i, A i ≤ c • B i) : tensor (fun i => root (A i)) * tensor (fun i => support (B i)) = tensor (fun i => root (A i)) := by rw [← tensor_mul] congr 1 funext i exact root_mul_support_of_le_smul _ _ (hB i) (hA i) c (hAB i) end SpinAngle.TensorMatrix namespace SpinAngle open scoped BigOperators variable {R C : Type*} [contextInstance4250_23 : Fintype R] [contextInstance4250_35 : DecidableEq R] [contextInstance4250_51 : Fintype C] [contextInstance4250_63 : DecidableEq C] lemma sum_occupiedRow_card (W : Finset (R × C)) : ∑ i, (occupiedRow W i).card = W.card := by have he (i : R) : (occupiedRow W i).card = ∑ j : C, if (i,j) ∈ W then 1 else 0 := by simp only [occupiedRow, Finset.card_eq_sum_ones, Finset.sum_filter] simp only [he] rw [← Finset.sum_product (f := fun x : R × C => if x ∈ W then (1 : ℕ) else 0)] simp only [Finset.univ_product_univ, ← Finset.sum_filter] have he : (Finset.univ.filter fun x : R × C => x ∈ W) = W := by ext x; simp rw [he] simp lemma sum_missingInRow (W : Finset (R × C)) : ∑ i, missingInRow W i = Fintype.card R * Fintype.card C - W.card := by have he : (∑ i, missingInRow W i) + W.card = Fintype.card R * Fintype.card C := by rw [← sum_occupiedRow_card W, ← Finset.sum_add_distrib] have hi (i : R) : missingInRow W i + (occupiedRow W i).card = Fintype.card C := Nat.sub_add_cancel (occupiedRow_card_le W i) simp [hi] omega lemma boardPhi_nonneg (W : Finset (R × C)) : 0 ≤ missingCellProductFactor (Fintype.card C) (missingInRow W) := Finset.prod_nonneg fun rowIndex _ => missingCellFactor_nonneg _ (missingInRow W rowIndex) lemma boardPhi_le_exp_holes (W : Finset (R × C)) : missingCellProductFactor (Fintype.card C) (missingInRow W) ≤ Real.exp (Fintype.card R * Fintype.card C - W.card : ℕ) := by have h := missingCellProduct (Fintype.card C) (missingInRow W) (missingInRow_le W) have hs : (∑ i, (missingInRow W i : ℝ)) = (Fintype.card R * Fintype.card C - W.card : ℕ) := by exact_mod_cast sum_missingInRow W rwa [hs] at h end SpinAngle namespace SpinAngle open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder open SpinAngle.Young structure TypeLabel (q : ℕ) where even : YoungDiagram odd : YoungDiagram even_height : even.colLen 0 ≤ q odd_height : odd.colLen 0 ≤ q namespace TypeLabel variable {q : ℕ} def size (D : TypeLabel q) : ℕ := Fintype.card (Cells D.even) + Fintype.card (Cells D.odd) def entropy (D : TypeLabel q) : ℝ := SignedCarrier.entropy D.even D.odd q def dimension (D : TypeLabel q) : ℕ := SignedCarrier.dimension D.even D.odd q D.even_height D.odd_height def projection (D : TypeLabel q) (τ : Type*) [Fintype τ] [DecidableEq τ] : Matrix (τ → SignedTensor.Spin q) (τ → SignedTensor.Spin q) ℂ := SignedCarrier.matrixProjection D.even D.odd q D.even_height D.odd_height lemma projection_isStarProjection (D : TypeLabel q) (τ : Type*) [Fintype τ] [DecidableEq τ] : IsStarProjection (D.projection τ) := SignedCarrier.matrixProjection_isStarProjection D.even D.odd q D.even_height D.odd_height lemma dimension_pos (D : TypeLabel q) : 0 < D.dimension := SignedCarrier.dimension_pos D.even D.odd q D.even_height D.odd_height lemma dimension_le (D : TypeLabel q) : D.dimension ≤ (D.size + 1) ^ (q*(q-1)) := SignedCarrier.dimension_le D.even D.odd q D.even_height D.odd_height lemma entropy_eq_zero_of_size_zero (D : TypeLabel q) (hD : D.size = 0) : D.entropy = 0 := by have hs : ∑ i, SignedCarrier.parts D.even D.odd q i = 0 := (SignedCarrier.sum_parts D.even D.odd q D.even_height D.odd_height).trans hD have hi : ∀ i, SignedCarrier.parts D.even D.odd q i = 0 := by intro i have h := Finset.single_le_sum (s := Finset.univ) (fun j _ => Nat.zero_le (SignedCarrier.parts D.even D.odd q j)) (Finset.mem_univ i) rw [hs] at h exact Nat.eq_zero_of_le_zero h simp [entropy, SignedCarrier.entropy, EntropyMonomial.entropy, hi] end TypeLabel abbrev EvenDensity (q : ℕ) := {A : SignedTensor.Mat q × SignedTensor.Mat q // SignedTensor.IsEvenDensity A} namespace EvenDensity variable {q : ℕ} def matrix (A : EvenDensity q) : Matrix (SignedTensor.Spin q) (SignedTensor.Spin q) ℂ := SignedTensor.block A.val lemma nonneg (A : EvenDensity q) : 0 ≤ A.matrix := (SignedTensor.block_posSemidef A.val A.property.1 A.property.2.1).nonneg lemma trace_eq_one (A : EvenDensity q) : A.matrix.trace = 1 := by rw [matrix, SignedTensor.block_trace] exact A.property.2.2 def toDensity (A : EvenDensity q) : MatrixAnalysis.Density (SignedTensor.Spin q) := ⟨A.matrix,A.nonneg,A.trace_eq_one⟩ variable {J : Type*} [contextInstance4345_21 : Fintype J] [contextInstance4345_33 : Nonempty J] def mean (A : J → EvenDensity q) : EvenDensity q := ⟨SignedTensor.evenAverage (fun j => (A j).val), SignedTensor.evenAverage_isEvenDensity _ (fun j => (A j).property)⟩ lemma mean_matrix (A : J → EvenDensity q) [DecidableEq J] : (mean A).matrix = MatrixAnalysis.average (fun j => (A j).toDensity) := by unfold mean matrix SignedTensor.evenAverage rw [SignedTensor.block_real_smul, SignedTensor.block_sum] rfl end EvenDensity namespace TypeLabel variable {q : ℕ} variable {τ : Type*} [contextInstance4358_21 : Fintype τ] [contextInstance4358_33 : DecidableEq τ] lemma density_upper (D : TypeLabel q) (A : EvenDensity q) : ‖TensorMatrix.tensor (fun _ : τ => A.matrix) * D.projection τ‖ ≤ Real.exp (-D.entropy) := SignedCarrier.matrix_density_upper D.even D.odd q D.even_height D.odd_height A.val A.property lemma density_commute (D : TypeLabel q) (A : EvenDensity q) : Commute (TensorMatrix.tensor (fun _ : τ => A.matrix)) (D.projection τ) := SignedCarrier.matrix_density_commute D.even D.odd q D.even_height D.odd_height A.val A.property variable [contextInstance4367_9 : Nonempty (Fin q)] lemma density_mixture (D : TypeLabel q) : ∃ (ι : Type) (_ : Fintype ι) (w : ι → ℝ) (R : ι → EvenDensity q), (∀ i, 0 ≤ w i) ∧ (∑ i, w i = 1) ∧ D.projection τ ≤ (Real.exp D.entropy * (D.dimension : ℝ)) • ∑ i, w i • TensorMatrix.tensor (fun _ : τ => (R i).matrix) := by obtain ⟨ι,hι,w,R,hw,hs,hR,hdom⟩ := SignedCarrier.exists_matrix_density_mixture (τ := τ) D.even D.odd q D.even_height D.odd_height let := hι exact ⟨ι,hι,w,fun i => ⟨R i,hR i⟩,hw,hs,hdom⟩ end TypeLabel namespace TypeBlocks universe u open MatrixAnalysis variable {q : ℕ} variable {τ : Type*} {J : Type u} [contextInstance4383_34 : Fintype τ] [contextInstance4383_46 : DecidableEq τ] [contextInstance4383_62 : Fintype J] [contextInstance4383_74 : DecidableEq J] abbrev Fiber (f : τ → J) (j : J) := BlockTensor.Fiber f j def projection (f : τ → J) (D : J → TypeLabel q) : Mat (τ → SignedTensor.Spin q) := BlockTensor.tensor f (fun j => (D j).projection (Fiber f j)) def entropy (D : J → TypeLabel q) : ℝ := ∑ j, (D j).entropy def dimension (D : J → TypeLabel q) : ℕ := ∏ j, (D j).dimension lemma projection_isStarProjection (f : τ → J) (D : J → TypeLabel q) : IsStarProjection (projection f D) := BlockTensor.tensor_projection f _ (fun j => (D j).projection_isStarProjection _) lemma density_commute (f : τ → J) (D : J → TypeLabel q) (A : EvenDensity q) : Commute (TensorMatrix.tensor (fun _ : τ => A.matrix)) (projection f D) := by have h := BlockTensor.tensor_commute f (fun j => TensorMatrix.tensor (fun _ : Fiber f j => A.matrix)) (fun j => (D j).projection (Fiber f j)) (fun j => (D j).density_commute A) have he : BlockTensor.tensor f (fun j => TensorMatrix.tensor (fun _ : Fiber f j => A.matrix)) = TensorMatrix.tensor (fun _ : τ => A.matrix) := by ext x y change (∏ j : J, ∏ t : Fiber f j, A.matrix (x t.val) (y t.val)) = ∏ t : τ, A.matrix (x t) (y t) exact BlockTensor.prod_fibers f (fun t => A.matrix (x t) (y t)) rwa [he] at h variable [contextInstance4410_9 : Nonempty (Fin q)] theorem density_mixture (f : τ → J) (D : J → TypeLabel q) : ∃ (ι : Type u) (_ : Fintype ι) (w : ι → ℝ) (R : ι → J → EvenDensity q), (∀ i, 0 ≤ w i) ∧ (∑ i, w i = Real.exp (entropy D) * (dimension D : ℝ)) ∧ projection f D ≤ ∑ i, w i • TensorMatrix.tensor (fun t : τ => (R i (f t)).matrix) := by classical choose κ hκ w R hw hs hdom using (fun j : J => (D j).density_mixture (τ := Fiber f j)) let : ∀ j, Fintype (κ j) := hκ let c (j : J) : ℝ := Real.exp (D j).entropy * (D j).dimension let z (j : J) (i : κ j) : ℝ := c j * w j i have hz (j : J) (i : κ j) : 0 ≤ z j i := mul_nonneg (mul_nonneg (Real.exp_pos _).le (Nat.cast_nonneg _)) (hw j i) have hsum (j : J) : ∑ i, z j i = c j := by simp only [z, ← Finset.mul_sum, hs, mul_one] have hlocal (j : J) : (D j).projection (Fiber f j) ≤ ∑ i, z j i • TensorMatrix.tensor (fun _ : Fiber f j => (R j i).matrix) := by simpa only [z, c, Finset.smul_sum, smul_smul] using hdom j refine ⟨(∀ j, κ j), inferInstance, (fun i => ∏ j, z j (i j)), (fun i j => R j (i j)), (fun i => Finset.prod_nonneg fun j _ => hz j (i j)), ?_, ?_⟩ · rw [← Fintype.prod_sum] simp only [hsum, c, Finset.prod_mul_distrib] rw [← Real.exp_sum] simp only [entropy, dimension, Nat.cast_prod] · have h := BlockTensor.tensor_mono f _ _ (fun j => ((D j).projection_isStarProjection (Fiber f j)).nonneg) hlocal rw [BlockTensor.tensor_sum_smul] at h have he (i : ∀ j, κ j) : BlockTensor.tensor f (fun j => TensorMatrix.tensor (fun _ : Fiber f j => (R j (i j)).matrix)) = TensorMatrix.tensor (fun t : τ => (R (f t) (i (f t))).matrix) := by ext x y change (∏ j : J, ∏ t : Fiber f j, (R j (i j)).matrix (x t.val) (y t.val)) = ∏ t : τ, (R (f t) (i (f t))).matrix (x t) (y t) calc _ = ∏ j : J, ∏ t : Fiber f j, (R (f t.val) (i (f t.val))).matrix (x t.val) (y t.val) := by apply Finset.prod_congr rfl intro j hj apply Finset.prod_congr rfl intro t ht rw [t.property] _ = _ := BlockTensor.prod_fibers f (fun t => (R (f t) (i (f t))).matrix (x t) (y t)) simpa only [he, projection] using h end TypeBlocks end SpinAngle end end end end end end end end end end end end end end end end end end end OAI