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import OAI.RepresentationTheory.Saxl.Main
import OAI.RepresentationTheory.UniversalSquare.Support.ProductInduction
import OAI.RepresentationTheory.UniversalSquare.Specht.CyclicRestriction
import OAI.RepresentationTheory.UniversalSquare.Support.CoindProduct
namespace OAI
/-! Packing Support. -/
section
noncomputable section
open scoped TensorProduct MonoidAlgebra
namespace Saxl
theorem cyclic_supportLE_coind {G H X Y : Type*}
[Group G] [Finite G] [Group H] [Finite H]
[AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y]
(ρ : Representation ℂ G X) (σ : Representation ℂ H Y)
(φ : H →* G) (v : X)
(T : Representation.IntertwiningMap σ (ρ.comp φ)) (hv : v ∈ T.range) :
SupportLE (cyclic ρ v).toRepresentation (Representation.coind φ σ) := by
apply intertwining_separates_of_simple_support
intro S hS
let := hS
have hi : subrepInclusion S ≠ 0 := by
intro hz
let : Nontrivial S.toRepresentation.asModule :=
IsSimpleModule.nontrivial ℂ[G] S.toRepresentation.asModule
let : Nontrivial S.toSubmodule := S.toRepresentation.asModuleEquiv.symm.toEquiv.nontrivial
obtain ⟨x,hx⟩ := exists_ne (0 : S.toSubmodule)
apply hx
apply Subtype.ext
exact congrArg (fun f : Representation.IntertwiningMap S.toRepresentation
(cyclic ρ v).toRepresentation => f x) hz
obtain ⟨F,hF⟩ := cyclic_restriction_support S.toRepresentation σ ρ φ v T hv
(subrepInclusion S) hi
exact ⟨coindLift φ F,coindLift_ne_zero φ F hF⟩
theorem SupportLE.cyclic_image {G X Y Z : Type*} [Group G] [Finite G]
[AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y]
[AddCommGroup Z] [Module ℂ Z]
{ρ : Representation ℂ G X} {σ : Representation ℂ G Y}
{τ : Representation ℂ G Z} (v : X)
(T : Representation.IntertwiningMap ρ σ) (h : SupportLE (cyclic ρ v).toRepresentation τ) :
SupportLE (cyclic σ (T v)).toRepresentation τ := by
let Q := T.comp (subrepInclusion (cyclic ρ v))
have hm : T v ∈ Q.range := ⟨⟨v,mem_cyclic ρ v⟩,rfl⟩
obtain ⟨i,hi⟩ := subrepresentation_embeds_of_le_range Q (cyclic σ (T v))
((cyclic_le _ _ _).mpr hm)
exact (SupportLE.of_injective i hi).trans h
def cyclicProductMap {n a b d : ℕ} (e : Fin n ≃ Fin a ⊕ Fin b)
(v : WordSpace a d) (u : WordSpace b d) :
Representation.IntertwiningMap
(outer (cyclic (wordRep a d) v).toRepresentation (cyclic (wordRep b d) u).toRepresentation)
((wordRep n d).comp (sumPermHom e)) :=
externalWordMap e (subrepInclusion (cyclic (wordRep a d) v))
(subrepInclusion (cyclic (wordRep b d) u))
lemma cyclicProduct_mem_range {n a b d : ℕ} (e : Fin n ≃ Fin a ⊕ Fin b)
(v : WordSpace a d) (u : WordSpace b d) :
positionProduct e v u ∈ (cyclicProductMap e v u).range := by
refine ⟨(⟨v,mem_cyclic _ _⟩ : (cyclic (wordRep a d) v).toSubmodule) ⊗ₜ[ℂ]
(⟨u,mem_cyclic _ _⟩ : (cyclic (wordRep b d) u).toSubmodule), ?_⟩
exact positionTensor_tmul e v u
theorem product_supportLE_coind {n a b d : ℕ} {G H X Y : Type*}
[Group G] [Finite G] [Group H] [Finite H]
[AddCommGroup X] [Module ℂ X] [FiniteDimensional ℂ X]
[AddCommGroup Y] [Module ℂ Y] [FiniteDimensional ℂ Y]
(φ : G →* Equiv.Perm (Fin a)) (ψ : H →* Equiv.Perm (Fin b))
(ρ : Representation ℂ G X) (σ : Representation ℂ H Y)
(e : Fin n ≃ Fin a ⊕ Fin b) (v : WordSpace a d) (u : WordSpace b d)
(hv : SupportLE (cyclic (wordRep a d) v).toRepresentation (Representation.coind φ ρ))
(hu : SupportLE (cyclic (wordRep b d) u).toRepresentation (Representation.coind ψ σ)) :
SupportLE (cyclic (wordRep n d) (positionProduct e v u)).toRepresentation
(Representation.coind ((sumPermHom e).comp (φ.prodMap ψ)) (outer ρ σ)) := by
classical
let : AddCommGroup ((cyclic (wordRep a d) v).toSubmodule ⊗[ℂ]
(cyclic (wordRep b d) u).toSubmodule) := Module.addCommMonoidToAddCommGroup ℂ
have h₀ := cyclic_supportLE_coind (wordRep n d)
(outer (cyclic (wordRep a d) v).toRepresentation (cyclic (wordRep b d) u).toRepresentation)
(sumPermHom e) (positionProduct e v u) (cyclicProductMap e v u) (cyclicProduct_mem_range e v u)
have h₁ := (hv.outer hu).coind (sumPermHom e)
have h₂ := (SupportLE.of_injective (coindOuter φ ψ ρ σ)
(coindOuter_injective φ ψ ρ σ)).coind (sumPermHom e)
have h₃ := SupportLE.of_injective (coindTrans (φ.prodMap ψ) (sumPermHom e) (outer ρ σ))
(coindTrans_injective (φ.prodMap ψ) (sumPermHom e) (outer ρ σ))
exact ((h₀.trans h₁).trans h₂).trans h₃
theorem young_cyclic_support {n : ℕ} (θ μ : YoungDiagram) (t : Tableau n μ)
(hc : μ.card = θ.card) (hd : Dominates μ θ)
(w : Fin n → Fin (θ.colLen 0)) (f : Equiv.Perm (Fin (θ.colLen 0)))
(hw : ∀ j, (Finset.univ.filter (fun i => w i = f j)).card = θ.rowLen j) :
∃ F : Representation.IntertwiningMap (spechtRep t)
(cyclic (wordRep n (θ.colLen 0)) (Pi.single w 1)).toRepresentation, F ≠ 0 := by
obtain ⟨F,hF⟩ := young_orbit_map θ μ t hc hd w f hw
let Q := F.comp (subrepInclusion (cyclic (wordRep n (θ.colLen 0)) (Pi.single w 1)))
have hQ : Q ≠ 0 := by
intro h0
exact hF (congrArg (fun T : Representation.IntertwiningMap
(cyclic (wordRep n (θ.colLen 0)) (Pi.single w 1)).toRepresentation (spechtRep t) =>
T ⟨Pi.single w 1,mem_cyclic _ _⟩) h0)
exact intertwining_reverse_nonzero Q hQ
end Saxl
end
end
end OAI
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