openAI-math-dataset / lean /OAI /RepresentationTheory /UniversalSquare /Support /CandidateSupport.lean
| import Mathlib | |
| import OAI.RepresentationTheory.Saxl.Main | |
| import OAI.RepresentationTheory.UniversalSquare.Support.TransposeSupport | |
| import OAI.RepresentationTheory.UniversalSquare.Support.CandidateCut | |
| namespace OAI | |
| /-! Candidate Support. -/ | |
| section | |
| noncomputable section | |
| open scoped TensorProduct | |
| namespace Saxl | |
| theorem kronecker_pos_of_tensor_cyclic_support {n : ℕ} {α β μ : YoungDiagram} | |
| (a a' : Tableau n α) (b b' : Tableau n β) (t : Tableau n μ) | |
| (f : Representation.IntertwiningMap (spechtRep t) | |
| (cyclic (wordRep n (α.colLen 0 * β.colLen 0)) | |
| (wordTensor _ _ _ (polytabloid a' ⊗ₜ[ℂ] polytabloid b'))).toRepresentation) | |
| (hf : f ≠ 0) : 0 < kronecker a b t := by | |
| let S := cyclic (wordRep n (α.colLen 0 * β.colLen 0)) | |
| (wordTensor _ _ _ (polytabloid a' ⊗ₜ[ℂ] polytabloid b')) | |
| 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 he := congrArg (fun H => H x) hz | |
| change F (f x) = 0 at he | |
| change F (f x) = F 0 | |
| simpa only [map_zero] using he | |
| lemma pairWord_mem_alphabetLeft_gen {n a b : ℕ} (A : Fin a → Prop) | |
| (x : WordSpace n a) (y : WordSpace n b) (hx : x ∈ alphabetSub n a A) : | |
| wordTensor n a b (x ⊗ₜ[ℂ] y) ∈ | |
| alphabetSub n (a*b) (fun c => A (finProdFinEquiv.symm c).1) := by | |
| apply (alphabetSub_iff _ _).mpr | |
| intro w hw i | |
| rw [wordTensor_tmul] at hw | |
| exact (alphabetSub_iff A x).mp hx (splitLeft w) (mul_ne_zero_iff.mp hw).1 i | |
| end Saxl | |
| namespace UniversalTensorSquare | |
| open Saxl | |
| theorem kronecker_pos_of_rowColumn_support {n : ℕ} {α μ : YoungDiagram} | |
| (hα : α.transpose = α) (a a' : Tableau n α) (t : Tableau n μ) | |
| (f : Representation.IntertwiningMap (spechtRep t) (rowColumnCyclic a').toRepresentation) | |
| (hf : f ≠ 0) : 0 < kronecker a a t := by | |
| have hp := kronecker_pos_of_tensor_cyclic_support | |
| (transposeTableau a) (transposeTableau a') a a' t f hf | |
| obtain ⟨g,hg⟩ := (kronecker_pos_iff _ _ _).mp hp | |
| let E := spechtShapeEquiv hα (transposeTableau a) a | |
| let F := E.toIntertwiningMap.tensor (Representation.IntertwiningMap.id (spechtRep a)) | |
| have hF : Function.Injective F := | |
| (TensorProduct.congr E.toLinearEquiv (LinearEquiv.refl ℂ (Specht a))).injective | |
| apply (kronecker_pos_iff _ _ _).mpr | |
| refine ⟨F.comp g, ?_⟩ | |
| intro hz | |
| apply hg | |
| apply Representation.IntertwiningMap.ext | |
| apply LinearMap.ext | |
| intro x | |
| apply hF | |
| have he := congrArg (fun H => H x) hz | |
| change F (g x) = 0 at he | |
| change F (g x) = F 0 | |
| simpa only [map_zero] using he | |
| lemma candidateHighLetter_injective (M b δ : ℕ) (hM : 4 ≤ M) : | |
| Function.Injective (candidateHighLetter M b δ hM) := by | |
| intro a c hac | |
| have ha := a.isLt | |
| have hc := c.isLt | |
| simp only [staircase_colLen, Nat.sub_zero] at ha hc | |
| have he := congrArg Fin.val hac | |
| dsimp [candidateHighLetter] at he | |
| apply Fin.ext | |
| split_ifs at he <;> omega | |
| lemma candidateHighMap_injective {n : ℕ} (M b δ : ℕ) (hM : 4 ≤ M) : | |
| Function.Injective (candidateHighMap (n := n) M b δ hM) := by | |
| apply pairWordMap_injective | |
| · apply letterLift_injective | |
| exact (Fin.cast_injective _).comp (candidateHighLetter_injective M b δ hM) | |
| · exact letterLift_injective _ (candidateHighLetter_injective M b δ hM) | |
| lemma candidateHighLetter_marked (M b δ : ℕ) (hM : 4 ≤ M) | |
| (a : Fin ((staircase (M-2)).colLen 0)) : | |
| candidateMark M b (candidateHighLetter M b δ hM a).val := by | |
| have ha := a.isLt | |
| simp only [staircase_colLen, Nat.sub_zero] at ha | |
| dsimp [candidateMark, candidateHighLetter] | |
| split_ifs <;> omega | |
| lemma candidateHighWord_alphabet (M b δ : ℕ) (hM : 4 ≤ M) : | |
| candidateHighWord M b δ hM ∈ alphabetSub _ _ | |
| (fun a => candidateMark M b ((finProdFinEquiv.symm a).1).val) := by | |
| unfold candidateHighWord candidateHighMap staircaseWord | |
| rw [pairWordMap_tensor] | |
| apply pairWord_mem_alphabetLeft_gen (fun a => candidateMark M b a.val) | |
| apply letterLift_mem_alphabet | |
| intro a | |
| exact candidateHighLetter_marked M b δ hM a | |
| lemma candidateBandRow_alphabet {n M b δ : ℕ} (hM : 4 ≤ M) | |
| (t : Tableau n (candidate M b δ)) : | |
| candidateBandRow hM t ∈ alphabetSub _ _ (fun a => ¬ candidateMark M b a.val) := by | |
| apply altWord_mem_alphabet | |
| intro j | |
| have hn := candidateCutPositions_high t ((candidateCutPositions t).symm (Sum.inr j)) | |
| simp only [Equiv.apply_symm_apply, Sum.isLeft_inr, Bool.false_eq_true, false_iff] at hn | |
| have hr := candidateRotateRow_mark M b δ hM (candidateBandTableau t j) | |
| have hh : ¬candidateMark M b (candidateRotateRow M b δ hM (candidateBandTableau t j)).val.2 := | |
| fun hh => hn (hr.mp hh) | |
| simpa [rightWord, rowWord, transposeTableau, candidateRotateRowPositions, candidateBandTableau] | |
| using hh | |
| lemma candidateBandWord_alphabet {n M b δ : ℕ} (hM : 4 ≤ M) | |
| (t : Tableau n (candidate M b δ)) : | |
| candidateBandWord hM t ∈ alphabetSub _ _ | |
| (fun a => ¬candidateMark M b ((finProdFinEquiv.symm a).1).val) := by | |
| exact pairWord_mem_alphabetLeft_gen (fun a => ¬candidateMark M b a.val) _ _ | |
| (candidateBandRow_alphabet hM t) | |
| lemma candidateBandWord_ne_zero {n M b δ : ℕ} (hM : 4 ≤ M) | |
| (t : Tableau n (candidate M b δ)) : candidateBandWord hM t ≠ 0 := by | |
| intro hz | |
| apply candidate_projected_ne_zero hM t | |
| rw [candidate_cut_factor hM t, hz] | |
| ext w | |
| change _ * 0 = 0 | |
| exact mul_zero _ | |
| theorem candidateHighWord_support (M b δ : ℕ) (hM : 4 ≤ M) | |
| (μ : YoungDiagram) (t : Tableau (staircase (M-2)).card μ) : | |
| ∃ F : Representation.IntertwiningMap (spechtRep t) | |
| (cyclic (wordRep _ _) (candidateHighWord M b δ hM)).toRepresentation, F ≠ 0 := by | |
| obtain ⟨f,hf⟩ := full_cyclic_support (M-2) (by omega) μ t | |
| exact cyclic_support_map (candidateHighMap M b δ hM) | |
| (candidateHighMap_injective M b δ hM) (staircaseWord (M-2)) f hf | |
| theorem candidate_triangle_quotient {n M b δ : ℕ} (hM : 4 ≤ M) | |
| (t : Tableau n (candidate M b δ)) : | |
| ∃ F : Representation.IntertwiningMap (rowColumnCyclic t).toRepresentation | |
| (cyclic (wordRep _ _) | |
| (positionProduct (candidateCutPositions t) (candidateHighWord M b δ hM) | |
| (candidateBandWord hM t))).toRepresentation, Function.Surjective F := by | |
| rw [← candidate_cut_factor hM t] | |
| exact ⟨cyclicMap (sameMarkProjection _ _ _ (candidateMark M b)) (rowColumnWord t), | |
| cyclicMap_surjective _ _⟩ | |
| theorem candidate_cut_strip_support {n M b δ : ℕ} (hM : 4 ≤ M) | |
| (a : Tableau n (candidate M b δ)) | |
| (ν μ : YoungDiagram) (hν : ν.card = (staircase (M-2)).card) | |
| (t : Tableau n μ) (bs : List ℕ) (hs : SizedStripChain bs ν μ) | |
| (hband : ∀ (η : YoungDiagram) (r : Tableau (candidateBandSize a) η), | |
| η.colLen 0 ≤ bs.length → | |
| ∃ F : Representation.IntertwiningMap (spechtRep r) | |
| (cyclic (wordRep _ _) (candidateBandWord hM a)).toRepresentation, F ≠ 0) : | |
| ∃ F : Representation.IntertwiningMap (spechtRep t) (rowColumnCyclic a).toRepresentation, | |
| F ≠ 0 := by | |
| obtain ⟨f,hf⟩ := candidateHighWord_support M b δ hM ν (canonicalTableau ν hν) | |
| have H := strip_product_support hs (canonicalTableau ν hν) t (candidateCutPositions a) | |
| (candidateHighWord M b δ hM) (candidateBandWord hM a) | |
| (fun α => candidateMark M b ((finProdFinEquiv.symm α).1).val) | |
| (candidateHighWord_alphabet M b δ hM) (candidateBandWord_alphabet hM a) f hf hband | |
| rw [← candidate_cut_factor hM a] at H | |
| obtain ⟨F,hF⟩ := H | |
| exact cyclic_projection_support (sameMarkProjection _ _ _ (candidateMark M b)) | |
| (rowColumnWord a) F hF | |
| theorem candidate_kronecker_pos_of_strip_band {n M b δ : ℕ} (hM : 4 ≤ M) | |
| (a : Tableau n (candidate M b δ)) | |
| (ν μ : YoungDiagram) (hν : ν.card = (staircase (M-2)).card) | |
| (t : Tableau n μ) (bs : List ℕ) (hs : SizedStripChain bs ν μ) | |
| (hband : ∀ (η : YoungDiagram) (r : Tableau (candidateBandSize a) η), | |
| η.colLen 0 ≤ bs.length → | |
| ∃ F : Representation.IntertwiningMap (spechtRep r) | |
| (cyclic (wordRep _ _) (candidateBandWord hM a)).toRepresentation, F ≠ 0) : | |
| 0 < kronecker a a t := by | |
| obtain ⟨F,hF⟩ := candidate_cut_strip_support hM a ν μ hν t bs hs hband | |
| exact kronecker_pos_of_rowColumn_support (candidate_transpose M b δ) a a t F hF | |
| end UniversalTensorSquare | |
| end | |
| end | |
| end OAI | |