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import OAI.RepresentationTheory.Saxl.Main
import OAI.RepresentationTheory.UniversalSquare.Support.CoordinateLine
import OAI.RepresentationTheory.UniversalSquare.Support.DualTransfer
namespace OAI
/-! Projected Tensor. -/
section
noncomputable section
open scoped TensorProduct
namespace Saxl
def invariantCoordinateProjection {n d : ℕ}
(P : (Fin n → Fin d) → Prop)
(hP : ∀ (g : Equiv.Perm (Fin n)) w, P (w ∘ g) ↔ P w) :
Representation.IntertwiningMap (wordRep n d) (wordRep n d) where
toLinearMap := coordinateProjection P
isIntertwining' g := by
classical
apply LinearMap.ext
intro x
funext w
change (if P w then x (w ∘ g) else 0) = (if P (w ∘ g) then x (w ∘ g) else 0)
rw [hP]
def projectedSpechtTensor {n : ℕ} {α β : YoungDiagram}
(a : Tableau n α) (b : Tableau n β)
(P : (Fin n → Fin (α.colLen 0 * β.colLen 0)) → Prop)
(hP : ∀ (g : Equiv.Perm (Fin n)) w, P (w ∘ g) ↔ P w) :=
((invariantCoordinateProjection P hP).comp (spechtTensorMap a b)).range
def projectedTensorQuotient {n : ℕ} {α β : YoungDiagram}
(a : Tableau n α) (b : Tableau n β)
(P : (Fin n → Fin (α.colLen 0 * β.colLen 0)) → Prop)
(hP : ∀ (g : Equiv.Perm (Fin n)) w, P (w ∘ g) ↔ P w) :
Representation.IntertwiningMap ((spechtRep a).tprod (spechtRep b))
(projectedSpechtTensor a b P hP).toRepresentation where
toLinearMap := ((invariantCoordinateProjection P hP).comp
(spechtTensorMap a b)).toLinearMap.rangeRestrict
isIntertwining' g := by
apply LinearMap.ext
intro x
apply Subtype.ext
exact LinearMap.congr_fun
(((invariantCoordinateProjection P hP).comp (spechtTensorMap a b)).isIntertwining' g) x
def projectedTensorEvaluation {n : ℕ} {α β : YoungDiagram}
(a : Tableau n α) (b : Tableau n β)
(P : (Fin n → Fin (α.colLen 0 * β.colLen 0)) → Prop)
(hP : ∀ (g : Equiv.Perm (Fin n)) w, P (w ∘ g) ↔ P w)
(w : Fin n → Fin (α.colLen 0 * β.colLen 0)) :
Module.Dual ℂ (projectedSpechtTensor a b P hP).toSubmodule :=
{ toFun := fun x => x.val w
map_add' := fun _ _ => rfl
map_smul' := fun _ _ => rfl }
lemma projectedTensorEvaluation_pullback {n : ℕ} {α β : YoungDiagram}
(a : Tableau n α) (b : Tableau n β)
(P : (Fin n → Fin (α.colLen 0 * β.colLen 0)) → Prop)
(hP : ∀ (g : Equiv.Perm (Fin n)) w, P (w ∘ g) ↔ P w)
(v : Fin n → Fin (α.colLen 0)) (w : Fin n → Fin (β.colLen 0))
(hp : P (mergeWords v w)) :
intertwiningDual (X := Specht a ⊗[ℂ] Specht b) (projectedTensorQuotient a b P hP)
(projectedTensorEvaluation a b P hP (mergeWords v w)) =
tensorDualEquiv (spechtRep a) (spechtRep b)
(coordinateRestriction a v ⊗ₜ[ℂ] coordinateRestriction b w) := by
classical
ext x y
change coordinateProjection P (spechtTensorMap a b (x ⊗ₜ[ℂ] y)) (mergeWords v w) = y.val w * x.val v
rw [coordinateProjection_apply, ite_eq_left hp]
rw [spechtTensorMap]
change wordTensor _ _ _ (x.val ⊗ₜ[ℂ] y.val) (mergeWords v w) = _
simp only [wordTensor_tmul, splitLeft_merge, splitRight_merge, mul_comm]
def dualSignMap {n : ℕ} {α : YoungDiagram} (a : Tableau n α)
(φ : (spechtRep a).Equiv (signTwist (spechtRep (transposeTableau a)))) :
Representation.IntertwiningMap (spechtRep a).dual
(signTwist (spechtRep (transposeTableau a))) :=
φ.toIntertwiningMap.comp (spechtSelfDual a).symm.toIntertwiningMap
def dualTensorSignMap {n : ℕ} {α β : YoungDiagram}
(a : Tableau n α) (b : Tableau n β)
(φ : (spechtRep a).Equiv (signTwist (spechtRep (transposeTableau a))))
(ψ : (spechtRep b).Equiv (signTwist (spechtRep (transposeTableau b)))) :
Representation.IntertwiningMap ((spechtRep a).tprod (spechtRep b)).dual
(wordRep n (α.transpose.colLen 0 * β.transpose.colLen 0)) :=
(spechtTensorMap (transposeTableau a) (transposeTableau b)).comp
((tensorSignCancel (spechtRep (transposeTableau a))
(spechtRep (transposeTableau b))).toIntertwiningMap.comp
(((dualSignMap a φ).tensor (dualSignMap b ψ)).comp
(tensorDualEquiv (spechtRep a) (spechtRep b)).symm.toIntertwiningMap))
lemma dualTensorSignMap_coordinates {n : ℕ} {α β : YoungDiagram}
(a s : Tableau n α) (b t : Tableau n β)
(φ : (spechtRep a).Equiv (signTwist (spechtRep (transposeTableau a))))
(ψ : (spechtRep b).Equiv (signTwist (spechtRep (transposeTableau b)))) :
∃ c : ℂ, c ≠ 0 ∧
dualTensorSignMap a b φ ψ
(tensorDualEquiv (spechtRep a) (spechtRep b)
(coordinateRestriction a (rowWord s) ⊗ₜ[ℂ] coordinateRestriction b (rowWord t))) =
c • wordTensor _ _ _
(polytabloid (transposeTableau s) ⊗ₜ[ℂ] polytabloid (transposeTableau t)) := by
obtain ⟨c,hc,hφ⟩ := signEquiv_coordinate_line a s φ
obtain ⟨d,hd,hψ⟩ := signEquiv_coordinate_line b t ψ
refine ⟨c*d, mul_ne_zero hc hd, ?_⟩
change spechtTensorMap (transposeTableau a) (transposeTableau b)
(TensorProduct.map (dualSignMap a φ).toLinearMap (dualSignMap b ψ).toLinearMap
((tensorDualEquiv (spechtRep a) (spechtRep b)).symm
(tensorDualEquiv (spechtRep a) (spechtRep b)
(coordinateRestriction a (rowWord s) ⊗ₜ[ℂ] coordinateRestriction b (rowWord t))))) = _
rw [Representation.Equiv.symm_apply_apply, TensorProduct.map_tmul]
rw [spechtTensorMap]
change wordTensor _ _ _
((φ (projectedCoordinate a (rowWord s))).val ⊗ₜ[ℂ]
(ψ (projectedCoordinate b (rowWord t))).val) = _
rw [hφ,hψ,TensorProduct.smul_tmul_smul,map_smul]
theorem pair_coordinate_in_projected_dual_image {n : ℕ} {α β : YoungDiagram}
(a s : Tableau n α) (b t : Tableau n β)
(P : (Fin n → Fin (α.colLen 0 * β.colLen 0)) → Prop)
(hP : ∀ (g : Equiv.Perm (Fin n)) w, P (w ∘ g) ↔ P w)
(hp : P (mergeWords (rowWord s) (rowWord t)))
(φ : (spechtRep a).Equiv (signTwist (spechtRep (transposeTableau a))))
(ψ : (spechtRep b).Equiv (signTwist (spechtRep (transposeTableau b)))) :
wordTensor _ _ _ (polytabloid (transposeTableau s) ⊗ₜ[ℂ]
polytabloid (transposeTableau t)) ∈
((dualTensorSignMap a b φ ψ).comp
(intertwiningDual (X := Specht a ⊗[ℂ] Specht b) (projectedTensorQuotient a b P hP))).range := by
obtain ⟨c,hc,he⟩ := dualTensorSignMap_coordinates a s b t φ ψ
apply (Submodule.smul_mem_iff _ hc).mp
refine ⟨projectedTensorEvaluation a b P hP (mergeWords (rowWord s) (rowWord t)), ?_⟩
change dualTensorSignMap a b φ ψ (intertwiningDual (X := Specht a ⊗[ℂ] Specht b) (projectedTensorQuotient a b P hP)
(projectedTensorEvaluation a b P hP (mergeWords (rowWord s) (rowWord t)))) = _
rw [projectedTensorEvaluation_pullback a b P hP _ _ hp]
exact he
theorem projectedTensor_support_of_coordinate {n : ℕ} {α β μ : YoungDiagram}
(a s : Tableau n α) (b t : Tableau n β) (u : Tableau n μ)
(P : (Fin n → Fin (α.colLen 0 * β.colLen 0)) → Prop)
(hP : ∀ (g : Equiv.Perm (Fin n)) w, P (w ∘ g) ↔ P w)
(hp : P (mergeWords (rowWord s) (rowWord t)))
(f : Representation.IntertwiningMap (spechtRep u)
(cyclic (wordRep n (α.transpose.colLen 0 * β.transpose.colLen 0))
(wordTensor _ _ _ (polytabloid (transposeTableau s) ⊗ₜ[ℂ]
polytabloid (transposeTableau t)))).toRepresentation) (hf : f ≠ 0) :
∃ F : Representation.IntertwiningMap (spechtRep u)
(projectedSpechtTensor a b P hP).toRepresentation, Function.Injective F := by
obtain ⟨φ⟩ := specht_sign_transpose a
obtain ⟨ψ⟩ := specht_sign_transpose b
let Q := (dualTensorSignMap a b φ ψ).comp
(intertwiningDual (X := Specht a ⊗[ℂ] Specht b) (projectedTensorQuotient a b P hP))
have hm := pair_coordinate_in_projected_dual_image a s b t P hP hp φ ψ
obtain ⟨i,hi⟩ := subrepresentation_embeds_of_le_range Q
(cyclic (wordRep n (α.transpose.colLen 0 * β.transpose.colLen 0))
(wordTensor _ _ _ (polytabloid (transposeTableau s) ⊗ₜ[ℂ]
polytabloid (transposeTableau t)))) ((cyclic_le _ _ _).mpr hm)
exact specht_support_of_dual u _ (i.comp f) (intertwining_comp_ne_zero i hi f hf)
end Saxl
end
end
end OAI
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