openAI-math-dataset / lean /OAI /RepresentationTheory /UniversalSquare /Contraction /GeneralDetection.lean
| import Mathlib | |
| import OAI.RepresentationTheory.Saxl.Main | |
| import OAI.RepresentationTheory.UniversalSquare.Specht.RepeatedColumns | |
| namespace OAI | |
| /-! General Detection. -/ | |
| section | |
| noncomputable section | |
| open scoped TensorProduct | |
| namespace Saxl | |
| lemma wordMap_pair_transpose {n d e : ℕ} (L : Fin d → Fin e → ℂ) | |
| (x : WordSpace n d) (y : WordSpace n e) : | |
| dotProduct (wordMap L x) y = dotProduct x (wordMap (fun b a => L a b) y) := by | |
| classical | |
| change (∑ b, (∑ a, x a * ∏ i, L (a i) (b i)) * y b) = | |
| ∑ a, x a * ∑ b, y b * ∏ i, L (a i) (b i) | |
| simp_rw [Finset.sum_mul, Finset.mul_sum] | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro a ha | |
| apply Finset.sum_congr rfl | |
| intro b hb | |
| ring | |
| theorem specht_to_cyclic_of_pair {n d : ℕ} {μ : YoungDiagram} | |
| (t : Tableau n μ) (L : Fin (μ.colLen 0) → Fin d → ℂ) | |
| (u : WordSpace n d) | |
| (h : dotProduct (wordMap L (polytabloid t)) u ≠ 0) : | |
| ∃ F : Representation.IntertwiningMap (spechtRep t) | |
| (cyclic (wordRep n d) u).toRepresentation, F ≠ 0 := by | |
| let pr := subrepProject (spechtSub t) | |
| (fun _ hx => cyclic_star _ (polytabloid_real t) hx) | |
| let F : Representation.IntertwiningMap | |
| (cyclic (wordRep n d) u).toRepresentation (spechtRep t) := | |
| pr.comp ((wordMap (fun b a => L a b)).comp (subrepInclusion _)) | |
| apply specht_map_back_of_nonzero _ t F | |
| intro hz | |
| have hp := subrepProject_pair (spechtSub t) | |
| (fun _ hx => cyclic_star _ (polytabloid_real t) hx) | |
| (wordMap (fun b a => L a b) u) ⟨polytabloid t, mem_cyclic _ _⟩ | |
| have he : pr (wordMap (fun b a => L a b) u) = 0 := | |
| congrArg (fun H : Representation.IntertwiningMap | |
| (cyclic (wordRep n d) u).toRepresentation (spechtRep t) => | |
| H ⟨u, mem_cyclic _ _⟩) hz | |
| rw [he] at hp | |
| change dotProduct (0 : WordSpace n (μ.colLen 0)) (polytabloid t) = | |
| dotProduct (wordMap (fun b a => L a b) u) (polytabloid t) at hp | |
| rw [zero_dotProduct] at hp | |
| rw [wordMap_pair_transpose, dotProduct_comm] | |
| at h | |
| exact h hp.symm | |
| end Saxl | |
| end | |
| end | |
| end OAI | |