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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
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