File size: 2,926 Bytes
607ee45 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import Mathlib
import OAI.RepresentationTheory.Saxl.Main
namespace OAI
/-! Irreducible embeddings and staircase tensor squares. -/
noncomputable section
open scoped TensorProduct
open scoped TensorProduct
universe u
namespace UniversalTensorSquares
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
end
end OAI
|