| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| open scoped BigOperators MonoidAlgebra Classical TensorProduct |
|
|
| namespace BinaryCoordinateSweeps.Irrep |
| open Representation |
|
|
| variable {G V W : Type*} [Group G] [Fintype G] |
| [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] |
| [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W] |
| (ρ : Representation ℂ G V) (σ : Representation ℂ G W) |
|
|
| omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in |
| lemma intertwiner_retraction (f : IntertwiningMap ρ σ) (hf : Function.Injective f) : |
| ∃ r : IntertwiningMap σ ρ, ∀ v, r (f v) = v := by |
| let fm := IntertwiningMap.equivLinearMapAsModule ρ σ f |
| have hm : LinearMap.ker fm = ⊥ := LinearMap.ker_eq_bot.mpr hf |
| have : NeZero (Nat.card G : ℂ) := ⟨by exact_mod_cast Nat.card_pos.ne'⟩ |
| obtain ⟨r,hr⟩ := MonoidAlgebra.exists_leftInverse_of_injective fm hm |
| refine ⟨(IntertwiningMap.equivLinearMapAsModule σ ρ).symm r, fun v => ?_⟩ |
| exact DFunLike.congr_fun hr v |
|
|
| theorem multiplicity_le_commutant [ρ.IsIrreducible] : |
| Module.finrank ℂ (IntertwiningMap ρ σ) ≤ Module.finrank ℂ (IntertwiningMap σ σ) := by |
| by_cases h : ∃ f : IntertwiningMap ρ σ, f ≠ 0 |
| · obtain ⟨f,hf⟩ := h |
| have hfi : Function.Injective f := (IsIrreducible.injective_or_eq_zero f).resolve_right hf |
| obtain ⟨r,hr⟩ := intertwiner_retraction ρ σ f hfi |
| let T : IntertwiningMap ρ σ →ₗ[ℂ] IntertwiningMap σ σ := |
| (IntertwiningMap.llcomp σ ρ σ).flip r |
| have hT : Function.Injective T := by |
| intro a b hab |
| ext v |
| have hh := congrArg (fun t : IntertwiningMap σ σ => t (f v)) hab |
| change a (r (f v)) = b (r (f v)) at hh |
| change a v = b v |
| simpa only [hr] using hh |
| exact LinearMap.finrank_le_finrank_of_injective hT |
| · have hs : Subsingleton (IntertwiningMap ρ σ) := ⟨by |
| intro a b |
| have ha : a=0 := by_contra (fun ha => h ⟨a,ha⟩) |
| have hb : b=0 := by_contra (fun hb => h ⟨b,hb⟩) |
| exact ha.trans hb.symm⟩ |
| rw [Module.finrank_eq_zero_of_subsingleton] |
| exact Nat.zero_le _ |
|
|
| def isotypicSpan : Submodule ℂ W := |
| ⨆ f : IntertwiningMap ρ σ, LinearMap.range f.toLinearMap |
|
|
| def evaluation : V ⊗[ℂ] IntertwiningMap ρ σ →ₗ[ℂ] W := |
| TensorProduct.lift { |
| toFun := fun v => { |
| toFun := fun f => f v |
| map_add' := fun _ _ => rfl |
| map_smul' := fun _ _ => rfl } |
| map_add' := by intro u v; ext f; exact map_add f u v |
| map_smul' := by intro c v; ext f; exact map_smul f c v } |
|
|
| omit [Fintype G] [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in |
| lemma evaluation_range : LinearMap.range (evaluation ρ σ) = isotypicSpan ρ σ := by |
| apply le_antisymm |
| · rintro _ ⟨t,rfl⟩ |
| induction t using TensorProduct.inductionOn with |
| | tmul v f => |
| exact (le_iSup (fun f : IntertwiningMap ρ σ => LinearMap.range f.toLinearMap) f) ⟨v,rfl⟩ |
| | add t u ht hu => simpa only [map_add] using Submodule.add_mem _ ht hu |
| · apply iSup_le |
| intro f w hw |
| obtain ⟨v,rfl⟩ := hw |
| exact ⟨TensorProduct.tmul ℂ v f, rfl⟩ |
|
|
| omit [Fintype G] in |
| lemma isotypicSpan_finrank : Module.finrank ℂ (isotypicSpan ρ σ) ≤ |
| Module.finrank ℂ V * Module.finrank ℂ (IntertwiningMap ρ σ) := by |
| rw [← evaluation_range] |
| simpa only [Module.finrank_tensorProduct] using LinearMap.finrank_range_le (evaluation ρ σ) |
|
|
| omit [Fintype G] [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in |
| lemma isotypicSpan_stable (g : G) (w : W) (hw : w ∈ isotypicSpan ρ σ) : |
| σ g w ∈ isotypicSpan ρ σ := by |
| apply Submodule.iSup_induction (fun f : IntertwiningMap ρ σ => LinearMap.range f.toLinearMap) |
| (motive := fun w => σ g w ∈ isotypicSpan ρ σ) hw |
| · intro f x hx |
| obtain ⟨v,rfl⟩ := hx |
| change σ g (f v) ∈ isotypicSpan ρ σ |
| rw [← IntertwiningMap.isIntertwining ρ σ f g v] |
| exact (le_iSup (fun f : IntertwiningMap ρ σ => LinearMap.range f.toLinearMap) f) |
| ⟨ρ g v,rfl⟩ |
| · simpa only [map_zero] using (isotypicSpan ρ σ).zero_mem |
| · intro x y hx hy |
| simpa only [map_add] using (isotypicSpan ρ σ).add_mem hx hy |
|
|
| omit [Fintype G] [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in |
| lemma isotypicSpan_map (g : G) : (isotypicSpan ρ σ).map (σ g) = isotypicSpan ρ σ := by |
| apply le_antisymm |
| · rintro _ ⟨w,hw,rfl⟩ |
| exact isotypicSpan_stable ρ σ g w hw |
| · intro w hw |
| refine ⟨σ g⁻¹ w, isotypicSpan_stable ρ σ g⁻¹ w hw, ?_⟩ |
| simp only [← Module.End.mul_apply, ← map_mul, mul_inv_cancel, map_one, Module.End.one_apply] |
|
|
| end BinaryCoordinateSweeps.Irrep |
|
|
| end |
|
|
| open scoped BigOperators Classical TensorProduct |
|
|
| namespace BinaryCoordinateSweeps.Irrep |
| open Representation |
|
|
| variable {G V W : Type*} [Group G] [Fintype G] |
| [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] |
| [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W] |
| (ρ : Representation ℂ G V) (σ : Representation ℂ G W) [ρ.IsIrreducible] |
|
|
| def schurScalar : IntertwiningMap ρ ρ ≃ₗ[ℂ] ℂ := |
| (LinearEquiv.ofBijective (Algebra.linearMap ℂ (IntertwiningMap ρ ρ)) |
| IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed).symm |
|
|
| omit [Fintype G] in |
| lemma schurScalar_smul (f : IntertwiningMap ρ ρ) (v : V) : |
| schurScalar ρ f • v = f v := by |
| have he := (schurScalar ρ).symm_apply_apply f |
| exact congrArg (fun t : IntertwiningMap ρ ρ => t v) he |
|
|
| omit [Fintype G] in |
| lemma schurScalar_one : schurScalar ρ (1 : IntertwiningMap ρ ρ) = 1 := by |
| apply (schurScalar ρ).symm.injective |
| rw [LinearEquiv.symm_apply_apply] |
| change (1 : IntertwiningMap ρ ρ) = algebraMap ℂ (IntertwiningMap ρ ρ) 1 |
| exact (map_one _).symm |
|
|
| def intertwinerPairing : IntertwiningMap ρ σ →ₗ[ℂ] |
| IntertwiningMap σ ρ →ₗ[ℂ] ℂ := |
| (LinearMap.compr₂ ((IntertwiningMap.llcomp ρ σ ρ).flip) (schurScalar ρ).toLinearMap) |
|
|
| omit [FiniteDimensional ℂ W] in |
| lemma intertwinerPairing_injective : Function.Injective (intertwinerPairing ρ σ) := by |
| apply (LinearMap.ker_eq_bot).mp |
| apply le_antisymm _ bot_le |
| intro f hf |
| change f = 0 |
| by_contra hn |
| have hfi := (IsIrreducible.injective_or_eq_zero f).resolve_right hn |
| obtain ⟨r,hr⟩ := intertwiner_retraction ρ σ f hfi |
| have hc : r.comp f = 1 := by ext v; exact hr v |
| have hz : intertwinerPairing ρ σ f = 0 := hf |
| have he := DFunLike.congr_fun hz r |
| change schurScalar ρ (r.comp f) = 0 at he |
| rw [hc,schurScalar_one] at he |
| exact one_ne_zero he |
|
|
| lemma exists_intertwiner_dual (l : Module.Dual ℂ (IntertwiningMap ρ σ)) : |
| ∃ r : IntertwiningMap σ ρ, ∀ f : IntertwiningMap ρ σ, ∀ v : V, |
| r (f v) = l f • v := by |
| have hs := (LinearMap.flip_surjective_iff₁ (B := intertwinerPairing ρ σ)).mpr |
| (intertwinerPairing_injective ρ σ) |
| obtain ⟨r,hr⟩ := hs l |
| refine ⟨r,fun f v => ?_⟩ |
| have he := DFunLike.congr_fun hr f |
| change schurScalar ρ (r.comp f) = l f at he |
| rw [← he] |
| exact (schurScalar_smul ρ (r.comp f) v).symm |
|
|
| theorem evaluation_injective : Function.Injective (evaluation ρ σ) := by |
| let b := Module.finBasis ℂ (IntertwiningMap ρ σ) |
| let e := TensorProduct.equivFinsuppOfBasisRight (M := V) b |
| apply (LinearMap.ker_eq_bot).mp |
| apply le_antisymm _ bot_le |
| intro t ht |
| change t = 0 |
| have ht0 : evaluation ρ σ t = 0 := ht |
| apply e.injective |
| apply Finsupp.ext |
| intro i |
| obtain ⟨r,hr⟩ := exists_intertwiner_dual ρ σ (b.coord i) |
| have he : ∀ x : V ⊗[ℂ] IntertwiningMap ρ σ, |
| r (evaluation ρ σ x) = e x i := by |
| intro x |
| induction x using TensorProduct.inductionOn with |
| | tmul v f => |
| change r (f v) = _ |
| rw [hr] |
| exact (TensorProduct.equivFinsuppOfBasisRight_apply_tmul_apply b v f i).symm |
| | add x y hx hy => simp only [map_add,Finsupp.add_apply,hx,hy] |
| simpa only [ht0,map_zero,Finsupp.zero_apply] using (he t).symm |
|
|
| lemma isotypicSpan_finrank_eq : Module.finrank ℂ (isotypicSpan ρ σ) = |
| Module.finrank ℂ V * Module.finrank ℂ (IntertwiningMap ρ σ) := by |
| rw [← evaluation_range, LinearMap.finrank_range_of_inj (evaluation_injective ρ σ), |
| Module.finrank_tensorProduct] |
|
|
| end BinaryCoordinateSweeps.Irrep |
|
|
| end |
|
|
| end OAI |
|
|
