| import Mathlib |
| import OAI.RepresentationTheory.Saxl.Main |
|
|
| namespace OAI |
|
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| /-! Irreducible embeddings and staircase tensor squares. -/ |
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|
| noncomputable section |
| open scoped TensorProduct |
| open scoped TensorProduct |
| universe u |
|
|
| namespace UniversalTensorSquares |
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|
| theorem contains_every_irreducible_of_positive {n : ℕ} |
| (lam : YoungDiagram) (hlam : lam.card = n) |
| (h : ∀ (nu : YoungDiagram) (hnu : nu.card = n), |
| 0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam) |
| (Saxl.canonicalTableau lam hlam) (Saxl.canonicalTableau nu hnu)) |
| {V : Type u} [AddCommGroup V] [Module ℂ V] [Module.Finite ℂ V] |
| (ρ : Representation ℂ (Equiv.Perm (Fin n)) V) |
| [Representation.IsIrreducible ρ] : |
| ∃ F : Representation.IntertwiningMap ρ |
| ((Saxl.spechtRep (Saxl.canonicalTableau lam hlam)).tprod |
| (Saxl.spechtRep (Saxl.canonicalTableau lam hlam))), Function.Injective F := by |
| obtain ⟨p, ⟨e⟩⟩ := Saxl.irreducible_equiv_specht ρ |
| obtain ⟨f, hf⟩ := (Saxl.kronecker_pos_iff (Saxl.canonicalTableau lam hlam) |
| (Saxl.canonicalTableau lam hlam) (Saxl.partitionTableau p)).mp |
| (h (Saxl.partitionDiagram p) (Saxl.partitionDiagram_card p)) |
| let := Saxl.specht_irreducible (Saxl.partitionTableau p) |
| have hi : Function.Injective f := by |
| have hi' := @Representation.IsIrreducible.injective_or_eq_zero |
| (Equiv.Perm (Fin n)) ℂ (Saxl.Specht (Saxl.partitionTableau p)) |
| (Saxl.Specht (Saxl.canonicalTableau lam hlam) ⊗[ℂ] |
| Saxl.Specht (Saxl.canonicalTableau lam hlam)) _ _ _ _ _ _ |
| (Saxl.spechtRep (Saxl.partitionTableau p)) |
| ((Saxl.spechtRep (Saxl.canonicalTableau lam hlam)).tprod |
| (Saxl.spechtRep (Saxl.canonicalTableau lam hlam))) f inferInstance |
| exact hi'.resolve_right hf |
| exact ⟨f.comp e.toIntertwiningMap, hi.comp e.injective⟩ |
|
|
| theorem universal_tensor_square_triangular (m : ℕ) (hm : 1 ≤ m) : |
| ∃ (lam : YoungDiagram) (hlam : lam.card = m * (m + 1) / 2), |
| (∀ (nu : YoungDiagram) (hnu : nu.card = m * (m + 1) / 2), |
| 0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam) |
| (Saxl.canonicalTableau lam hlam) (Saxl.canonicalTableau nu hnu)) ∧ |
| Representation.IsIrreducible (Saxl.spechtRep (Saxl.canonicalTableau lam hlam)) ∧ |
| ∀ {V : Type u} [AddCommGroup V] [Module ℂ V] [Module.Finite ℂ V] |
| (ρ : Representation ℂ (Equiv.Perm (Fin (m * (m + 1) / 2))) V) |
| [Representation.IsIrreducible ρ], |
| ∃ F : Representation.IntertwiningMap ρ |
| ((Saxl.spechtRep (Saxl.canonicalTableau lam hlam)).tprod |
| (Saxl.spechtRep (Saxl.canonicalTableau lam hlam))), |
| Function.Injective F := by |
| rw [← Saxl.staircase_card m] |
| refine ⟨Saxl.staircase m, rfl, ?_, Saxl.specht_irreducible _, ?_⟩ |
| · exact Saxl.saxl_conjecture m hm |
| · intro V _ _ _ ρ _ |
| exact Saxl.saxl_contains_every_irreducible m hm ρ |
|
|
| end UniversalTensorSquares |
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| end |
|
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| end OAI |
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|
