openAI-math-dataset / lean /OAI /RepresentationTheory /PartialPermutation /SpechtClassification.lean
| import Mathlib | |
| import OAI.RepresentationTheory.PartialPermutation.Plancherel | |
| import OAI.RepresentationTheory.PartialPermutation.Equivalences | |
| import OAI.RepresentationTheory.PartialPermutation.SpechtDimension | |
| import OAI.RepresentationTheory.PartialPermutation.SpechtShapes | |
| import OAI.RepresentationTheory.PartialPermutation.ConjugacyCounts | |
| import OAI.RepresentationTheory.PartialPermutation.ClassCharacters | |
| namespace OAI | |
| section | |
| namespace PartialPermutation | |
| namespace SpechtActual | |
| noncomputable section | |
| open YoungTabloid DiagramCounting | |
| open scoped Classical | |
| def labels {n : ℕ} (μ : Shape n) : μ.1.cells ≃ Fin n := | |
| (Fintype.equivFin μ.1.cells).trans (finCongr (by simpa using μ.2)) | |
| def rep {n : ℕ} (μ : Shape n) := | |
| (spechtRep μ.1).comp (labels μ).symm.permCongrHom.toMonoidHom | |
| instance {n : ℕ} (μ : Shape n) : Representation.IsIrreducible (rep μ) := by | |
| unfold rep | |
| infer_instance | |
| def index {n : ℕ} (μ : Shape n) : IrreducibleIndex (Equiv.Perm (Fin n)) := | |
| (irreducibleRep_complete (rep μ)).choose | |
| lemma index_spec {n : ℕ} (μ : Shape n) : | |
| Nonempty (Representation.Equiv (irreducibleRep (index μ)) (rep μ)) := | |
| (irreducibleRep_complete (rep μ)).choose_spec.1 | |
| lemma rep_equiv_shape_eq {n : ℕ} (μ ν : Shape n) | |
| (E : Representation.Equiv (rep μ) (rep ν)) : μ=ν := by | |
| apply Subtype.ext | |
| let e : μ.1.cells ≃ ν.1.cells := (labels μ).trans (labels ν).symm | |
| apply specht_equiv_shape_eq μ.1 ν.1 e E.toLinearEquiv | |
| intro g x | |
| have hh := LinearMap.congr_fun (E.isIntertwining' ((labels μ).permCongrHom g)) x | |
| change E (spechtRep μ.1 ((labels μ).symm.permCongrHom ((labels μ).permCongrHom g)) x)= | |
| spechtRep ν.1 ((labels ν).symm.permCongrHom ((labels μ).permCongrHom g)) (E x) at hh | |
| have he : (labels μ).symm.permCongrHom ((labels μ).permCongrHom g)=g := | |
| (labels μ).permCongrHom.symm_apply_apply g | |
| rw [he] at hh | |
| exact hh | |
| lemma index_injective (n : ℕ) : Function.Injective (@index n) := by | |
| intro μ ν h | |
| obtain ⟨e⟩ := index_spec μ | |
| obtain ⟨f⟩ := index_spec ν | |
| apply rep_equiv_shape_eq μ ν | |
| exact e.symm.trans (h ▸ f) | |
| lemma actual_irreducible_card_le (n : ℕ) : | |
| Fintype.card (IrreducibleIndex (Equiv.Perm (Fin n))) ≤ Fintype.card (Shape n) := by | |
| exact (irreducible_family_card_le_conjClasses (fun c => irreducibleSpace c) | |
| (fun c => irreducibleRep c) (componentRep_equiv_iff (regularRep (Equiv.Perm (Fin n))))).trans | |
| (symmetric_conjClasses_card_le n) | |
| theorem index_bijective (n : ℕ) : Function.Bijective (@index n) := by | |
| have hc := actual_irreducible_card_le n | |
| have hi := index_injective n | |
| have he := le_antisymm hc (Fintype.card_le_of_injective _ hi) | |
| exact ⟨hi,(Fintype.bijective_iff_injective_and_card _).mpr ⟨hi,he.symm⟩ |>.surjective⟩ | |
| def indexEquiv (n : ℕ) : Shape n ≃ IrreducibleIndex (Equiv.Perm (Fin n)) := | |
| Equiv.ofBijective index (index_bijective n) | |
| lemma standardCount_le_degree {n : ℕ} (μ : Shape n) : | |
| Tableau.standardCount μ.1 ≤ irreducibleDegree (index μ) := by | |
| obtain ⟨e⟩ := index_spec μ | |
| have hd := e.toLinearEquiv.finrank_eq | |
| exact (standardCount_le_specht_degree μ.1).trans_eq hd.symm | |
| end | |
| end SpechtActual | |
| end PartialPermutation | |
| end | |
| end OAI | |