openAI-math-dataset / lean /OAI /RepresentationTheory /UniversalSquare /Contraction /SquareDetection.lean
| import Mathlib | |
| import OAI.RepresentationTheory.Saxl.Main | |
| namespace OAI | |
| /-! Square Detection. -/ | |
| section | |
| noncomputable section | |
| open scoped TensorProduct | |
| namespace Saxl | |
| def spechtTensorMap {n : ℕ} {α β : YoungDiagram} | |
| (a : Tableau n α) (b : Tableau n β) : | |
| Representation.IntertwiningMap ((spechtRep a).tprod (spechtRep b)) | |
| (wordRep n (α.colLen 0 * β.colLen 0)) := | |
| (wordTensorIntertwiner _ _ _).comp ((spechtInclusion a).tensor (spechtInclusion b)) | |
| lemma spechtTensorMap_injective {n : ℕ} {α β : YoungDiagram} | |
| (a : Tableau n α) (b : Tableau n β) : Function.Injective (spechtTensorMap a b) := by | |
| apply (wordTensor _ _ _).injective.comp | |
| exact TensorProduct.map_injective_of_flat_flat _ _ | |
| (spechtSub a).toSubmodule.subtype_injective (spechtSub b).toSubmodule.subtype_injective | |
| lemma polytabloidTensor_real {n : ℕ} {α β : YoungDiagram} | |
| (a : Tableau n α) (b : Tableau n β) : | |
| star (wordTensor _ _ _ (polytabloid a ⊗ₜ[ℂ] polytabloid b)) = | |
| wordTensor _ _ _ (polytabloid a ⊗ₜ[ℂ] polytabloid b) := by | |
| funext w | |
| simp only [Pi.star_apply, wordTensor_tmul, star_mul] | |
| have h₁ := congrFun (polytabloid_real a) (splitLeft w) | |
| have h₂ := congrFun (polytabloid_real b) (splitRight w) | |
| change star (polytabloid b (splitRight w)) * star (polytabloid a (splitLeft w)) = _ | |
| rw [show star (polytabloid a (splitLeft w)) = _ from h₁, | |
| show star (polytabloid b (splitRight w)) = _ from h₂] | |
| exact mul_comm _ _ | |
| lemma polytabloidTensor_cyclic_le_range {n : ℕ} {α β : YoungDiagram} | |
| (a a' : Tableau n α) (b b' : Tableau n β) : | |
| cyclic (wordRep n (α.colLen 0 * β.colLen 0)) | |
| (wordTensor _ _ _ (polytabloid a' ⊗ₜ[ℂ] polytabloid b')) ≤ | |
| (spechtTensorMap a b).range := by | |
| apply (cyclic_le _ _ _).mpr | |
| exact ⟨(⟨polytabloid a', polytabloid_mem_all a a'⟩ : Specht a) ⊗ₜ[ℂ] | |
| (⟨polytabloid b', polytabloid_mem_all b b'⟩ : Specht b), rfl⟩ | |
| attribute [irreducible] spechtTensorMap | |
| theorem kronecker_pos_of_contraction {n : ℕ} {α β μ : YoungDiagram} | |
| (a a' : Tableau n α) (b b' : Tableau n β) (t s : Tableau n μ) | |
| (L : Fin (μ.colLen 0) → Fin (α.colLen 0 * β.colLen 0) → ℂ) | |
| (h : dotProduct (wordMap L (polytabloid s)) | |
| (wordTensor _ _ _ (polytabloid a' ⊗ₜ[ℂ] polytabloid b')) ≠ 0) : | |
| 0 < kronecker a b t := by | |
| let z := wordTensor _ _ _ (polytabloid a' ⊗ₜ[ℂ] polytabloid b') | |
| let S := cyclic (wordRep n (α.colLen 0 * β.colLen 0)) z | |
| obtain ⟨f, hf⟩ := hom_to_cyclic_of_pair L (spechtSub t) | |
| ⟨polytabloid s, polytabloid_mem_all t s⟩ z (polytabloidTensor_real a' b') h | |
| let F := @intertwiningLiftSubrep (Equiv.Perm (Fin n)) | |
| (Specht a ⊗[ℂ] Specht b) (WordSpace n (α.colLen 0 * β.colLen 0)) | |
| _ _ _ _ _ ((spechtRep a).tprod (spechtRep b)) | |
| (wordRep n (α.colLen 0 * β.colLen 0)) | |
| (spechtTensorMap a b) (spechtTensorMap_injective a b) | |
| S (polytabloidTensor_cyclic_le_range a a' b b') | |
| have hF : Function.Injective F := @intertwiningLiftSubrep_injective (Equiv.Perm (Fin n)) | |
| (Specht a ⊗[ℂ] Specht b) (WordSpace n (α.colLen 0 * β.colLen 0)) | |
| _ _ _ _ _ ((spechtRep a).tprod (spechtRep b)) | |
| (wordRep n (α.colLen 0 * β.colLen 0)) | |
| (spechtTensorMap a b) (spechtTensorMap_injective a b) | |
| S (polytabloidTensor_cyclic_le_range a a' b b') | |
| apply (kronecker_pos_iff a b t).mpr | |
| refine ⟨F.comp f, ?_⟩ | |
| intro hz | |
| apply hf | |
| apply Representation.IntertwiningMap.ext | |
| apply LinearMap.ext | |
| intro x | |
| apply hF | |
| have hzero := congrArg (fun H => H x) hz | |
| change F (f x) = 0 at hzero | |
| change F (f x) = F 0 | |
| simpa only [map_zero] using hzero | |
| end Saxl | |
| end | |
| end | |
| end OAI | |