openAI-math-dataset / lean /OAI /RepresentationTheory /UniversalSquare /Support /ProjectedTensor.lean
| import Mathlib | |
| 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 | |