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import OAI.RepresentationTheory.Saxl.Main
import OAI.RepresentationTheory.UniversalSquare.Support.TransposeSupport
import OAI.RepresentationTheory.UniversalSquare.Support.CandidateCut
namespace OAI
/-! Candidate Support. -/
section
noncomputable section
open scoped TensorProduct
namespace Saxl
theorem kronecker_pos_of_tensor_cyclic_support {n : ℕ} {α β μ : YoungDiagram}
(a a' : Tableau n α) (b b' : Tableau n β) (t : Tableau n μ)
(f : Representation.IntertwiningMap (spechtRep t)
(cyclic (wordRep n (α.colLen 0 * β.colLen 0))
(wordTensor _ _ _ (polytabloid a' ⊗ₜ[ℂ] polytabloid b'))).toRepresentation)
(hf : f ≠ 0) : 0 < kronecker a b t := by
let S := cyclic (wordRep n (α.colLen 0 * β.colLen 0))
(wordTensor _ _ _ (polytabloid a' ⊗ₜ[ℂ] polytabloid b'))
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 he := congrArg (fun H => H x) hz
change F (f x) = 0 at he
change F (f x) = F 0
simpa only [map_zero] using he
lemma pairWord_mem_alphabetLeft_gen {n a b : ℕ} (A : Fin a → Prop)
(x : WordSpace n a) (y : WordSpace n b) (hx : x ∈ alphabetSub n a A) :
wordTensor n a b (x ⊗ₜ[ℂ] y) ∈
alphabetSub n (a*b) (fun c => A (finProdFinEquiv.symm c).1) := by
apply (alphabetSub_iff _ _).mpr
intro w hw i
rw [wordTensor_tmul] at hw
exact (alphabetSub_iff A x).mp hx (splitLeft w) (mul_ne_zero_iff.mp hw).1 i
end Saxl
namespace UniversalTensorSquare
open Saxl
theorem kronecker_pos_of_rowColumn_support {n : ℕ} {α μ : YoungDiagram}
(hα : α.transpose = α) (a a' : Tableau n α) (t : Tableau n μ)
(f : Representation.IntertwiningMap (spechtRep t) (rowColumnCyclic a').toRepresentation)
(hf : f ≠ 0) : 0 < kronecker a a t := by
have hp := kronecker_pos_of_tensor_cyclic_support
(transposeTableau a) (transposeTableau a') a a' t f hf
obtain ⟨g,hg⟩ := (kronecker_pos_iff _ _ _).mp hp
let E := spechtShapeEquiv hα (transposeTableau a) a
let F := E.toIntertwiningMap.tensor (Representation.IntertwiningMap.id (spechtRep a))
have hF : Function.Injective F :=
(TensorProduct.congr E.toLinearEquiv (LinearEquiv.refl ℂ (Specht a))).injective
apply (kronecker_pos_iff _ _ _).mpr
refine ⟨F.comp g, ?_⟩
intro hz
apply hg
apply Representation.IntertwiningMap.ext
apply LinearMap.ext
intro x
apply hF
have he := congrArg (fun H => H x) hz
change F (g x) = 0 at he
change F (g x) = F 0
simpa only [map_zero] using he
lemma candidateHighLetter_injective (M b δ : ℕ) (hM : 4 ≤ M) :
Function.Injective (candidateHighLetter M b δ hM) := by
intro a c hac
have ha := a.isLt
have hc := c.isLt
simp only [staircase_colLen, Nat.sub_zero] at ha hc
have he := congrArg Fin.val hac
dsimp [candidateHighLetter] at he
apply Fin.ext
split_ifs at he <;> omega
lemma candidateHighMap_injective {n : ℕ} (M b δ : ℕ) (hM : 4 ≤ M) :
Function.Injective (candidateHighMap (n := n) M b δ hM) := by
apply pairWordMap_injective
· apply letterLift_injective
exact (Fin.cast_injective _).comp (candidateHighLetter_injective M b δ hM)
· exact letterLift_injective _ (candidateHighLetter_injective M b δ hM)
lemma candidateHighLetter_marked (M b δ : ℕ) (hM : 4 ≤ M)
(a : Fin ((staircase (M-2)).colLen 0)) :
candidateMark M b (candidateHighLetter M b δ hM a).val := by
have ha := a.isLt
simp only [staircase_colLen, Nat.sub_zero] at ha
dsimp [candidateMark, candidateHighLetter]
split_ifs <;> omega
lemma candidateHighWord_alphabet (M b δ : ℕ) (hM : 4 ≤ M) :
candidateHighWord M b δ hM ∈ alphabetSub _ _
(fun a => candidateMark M b ((finProdFinEquiv.symm a).1).val) := by
unfold candidateHighWord candidateHighMap staircaseWord
rw [pairWordMap_tensor]
apply pairWord_mem_alphabetLeft_gen (fun a => candidateMark M b a.val)
apply letterLift_mem_alphabet
intro a
exact candidateHighLetter_marked M b δ hM a
lemma candidateBandRow_alphabet {n M b δ : ℕ} (hM : 4 ≤ M)
(t : Tableau n (candidate M b δ)) :
candidateBandRow hM t ∈ alphabetSub _ _ (fun a => ¬ candidateMark M b a.val) := by
apply altWord_mem_alphabet
intro j
have hn := candidateCutPositions_high t ((candidateCutPositions t).symm (Sum.inr j))
simp only [Equiv.apply_symm_apply, Sum.isLeft_inr, Bool.false_eq_true, false_iff] at hn
have hr := candidateRotateRow_mark M b δ hM (candidateBandTableau t j)
have hh : ¬candidateMark M b (candidateRotateRow M b δ hM (candidateBandTableau t j)).val.2 :=
fun hh => hn (hr.mp hh)
simpa [rightWord, rowWord, transposeTableau, candidateRotateRowPositions, candidateBandTableau]
using hh
lemma candidateBandWord_alphabet {n M b δ : ℕ} (hM : 4 ≤ M)
(t : Tableau n (candidate M b δ)) :
candidateBandWord hM t ∈ alphabetSub _ _
(fun a => ¬candidateMark M b ((finProdFinEquiv.symm a).1).val) := by
exact pairWord_mem_alphabetLeft_gen (fun a => ¬candidateMark M b a.val) _ _
(candidateBandRow_alphabet hM t)
lemma candidateBandWord_ne_zero {n M b δ : ℕ} (hM : 4 ≤ M)
(t : Tableau n (candidate M b δ)) : candidateBandWord hM t ≠ 0 := by
intro hz
apply candidate_projected_ne_zero hM t
rw [candidate_cut_factor hM t, hz]
ext w
change _ * 0 = 0
exact mul_zero _
theorem candidateHighWord_support (M b δ : ℕ) (hM : 4 ≤ M)
(μ : YoungDiagram) (t : Tableau (staircase (M-2)).card μ) :
∃ F : Representation.IntertwiningMap (spechtRep t)
(cyclic (wordRep _ _) (candidateHighWord M b δ hM)).toRepresentation, F ≠ 0 := by
obtain ⟨f,hf⟩ := full_cyclic_support (M-2) (by omega) μ t
exact cyclic_support_map (candidateHighMap M b δ hM)
(candidateHighMap_injective M b δ hM) (staircaseWord (M-2)) f hf
theorem candidate_triangle_quotient {n M b δ : ℕ} (hM : 4 ≤ M)
(t : Tableau n (candidate M b δ)) :
∃ F : Representation.IntertwiningMap (rowColumnCyclic t).toRepresentation
(cyclic (wordRep _ _)
(positionProduct (candidateCutPositions t) (candidateHighWord M b δ hM)
(candidateBandWord hM t))).toRepresentation, Function.Surjective F := by
rw [← candidate_cut_factor hM t]
exact ⟨cyclicMap (sameMarkProjection _ _ _ (candidateMark M b)) (rowColumnWord t),
cyclicMap_surjective _ _⟩
theorem candidate_cut_strip_support {n M b δ : ℕ} (hM : 4 ≤ M)
(a : Tableau n (candidate M b δ))
(ν μ : YoungDiagram) (hν : ν.card = (staircase (M-2)).card)
(t : Tableau n μ) (bs : List ℕ) (hs : SizedStripChain bs ν μ)
(hband : ∀ (η : YoungDiagram) (r : Tableau (candidateBandSize a) η),
η.colLen 0 ≤ bs.length →
∃ F : Representation.IntertwiningMap (spechtRep r)
(cyclic (wordRep _ _) (candidateBandWord hM a)).toRepresentation, F ≠ 0) :
∃ F : Representation.IntertwiningMap (spechtRep t) (rowColumnCyclic a).toRepresentation,
F ≠ 0 := by
obtain ⟨f,hf⟩ := candidateHighWord_support M b δ hM ν (canonicalTableau ν hν)
have H := strip_product_support hs (canonicalTableau ν hν) t (candidateCutPositions a)
(candidateHighWord M b δ hM) (candidateBandWord hM a)
(fun α => candidateMark M b ((finProdFinEquiv.symm α).1).val)
(candidateHighWord_alphabet M b δ hM) (candidateBandWord_alphabet hM a) f hf hband
rw [← candidate_cut_factor hM a] at H
obtain ⟨F,hF⟩ := H
exact cyclic_projection_support (sameMarkProjection _ _ _ (candidateMark M b))
(rowColumnWord a) F hF
theorem candidate_kronecker_pos_of_strip_band {n M b δ : ℕ} (hM : 4 ≤ M)
(a : Tableau n (candidate M b δ))
(ν μ : YoungDiagram) (hν : ν.card = (staircase (M-2)).card)
(t : Tableau n μ) (bs : List ℕ) (hs : SizedStripChain bs ν μ)
(hband : ∀ (η : YoungDiagram) (r : Tableau (candidateBandSize a) η),
η.colLen 0 ≤ bs.length →
∃ F : Representation.IntertwiningMap (spechtRep r)
(cyclic (wordRep _ _) (candidateBandWord hM a)).toRepresentation, F ≠ 0) :
0 < kronecker a a t := by
obtain ⟨F,hF⟩ := candidate_cut_strip_support hM a ν μ hν t bs hs hband
exact kronecker_pos_of_rowColumn_support (candidate_transpose M b δ) a a t F hF
end UniversalTensorSquare
end
end
end OAI
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