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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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | import Mathlib
import OAI.RepresentationTheory.Saxl.Main
namespace OAI
/-! Square Detection. -/
section
noncomputable section
open scoped TensorProduct
namespace Saxl
def spechtTensorMap {n : ℕ} {α β : YoungDiagram}
(a : Tableau n α) (b : Tableau n β) :
Representation.IntertwiningMap ((spechtRep a).tprod (spechtRep b))
(wordRep n (α.colLen 0 * β.colLen 0)) :=
(wordTensorIntertwiner _ _ _).comp ((spechtInclusion a).tensor (spechtInclusion b))
lemma spechtTensorMap_injective {n : ℕ} {α β : YoungDiagram}
(a : Tableau n α) (b : Tableau n β) : Function.Injective (spechtTensorMap a b) := by
apply (wordTensor _ _ _).injective.comp
exact TensorProduct.map_injective_of_flat_flat _ _
(spechtSub a).toSubmodule.subtype_injective (spechtSub b).toSubmodule.subtype_injective
lemma polytabloidTensor_real {n : ℕ} {α β : YoungDiagram}
(a : Tableau n α) (b : Tableau n β) :
star (wordTensor _ _ _ (polytabloid a ⊗ₜ[ℂ] polytabloid b)) =
wordTensor _ _ _ (polytabloid a ⊗ₜ[ℂ] polytabloid b) := by
funext w
simp only [Pi.star_apply, wordTensor_tmul, star_mul]
have h₁ := congrFun (polytabloid_real a) (splitLeft w)
have h₂ := congrFun (polytabloid_real b) (splitRight w)
change star (polytabloid b (splitRight w)) * star (polytabloid a (splitLeft w)) = _
rw [show star (polytabloid a (splitLeft w)) = _ from h₁,
show star (polytabloid b (splitRight w)) = _ from h₂]
exact mul_comm _ _
lemma polytabloidTensor_cyclic_le_range {n : ℕ} {α β : YoungDiagram}
(a a' : Tableau n α) (b b' : Tableau n β) :
cyclic (wordRep n (α.colLen 0 * β.colLen 0))
(wordTensor _ _ _ (polytabloid a' ⊗ₜ[ℂ] polytabloid b')) ≤
(spechtTensorMap a b).range := by
apply (cyclic_le _ _ _).mpr
exact ⟨(⟨polytabloid a', polytabloid_mem_all a a'⟩ : Specht a) ⊗ₜ[ℂ]
(⟨polytabloid b', polytabloid_mem_all b b'⟩ : Specht b), rfl⟩
attribute [irreducible] spechtTensorMap
theorem kronecker_pos_of_contraction {n : ℕ} {α β μ : YoungDiagram}
(a a' : Tableau n α) (b b' : Tableau n β) (t s : Tableau n μ)
(L : Fin (μ.colLen 0) → Fin (α.colLen 0 * β.colLen 0) → ℂ)
(h : dotProduct (wordMap L (polytabloid s))
(wordTensor _ _ _ (polytabloid a' ⊗ₜ[ℂ] polytabloid b')) ≠ 0) :
0 < kronecker a b t := by
let z := wordTensor _ _ _ (polytabloid a' ⊗ₜ[ℂ] polytabloid b')
let S := cyclic (wordRep n (α.colLen 0 * β.colLen 0)) z
obtain ⟨f, hf⟩ := hom_to_cyclic_of_pair L (spechtSub t)
⟨polytabloid s, polytabloid_mem_all t s⟩ z (polytabloidTensor_real a' b') h
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 hzero := congrArg (fun H => H x) hz
change F (f x) = 0 at hzero
change F (f x) = F 0
simpa only [map_zero] using hzero
end Saxl
end
end
end OAI
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