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import OAI.RepresentationTheory.Saxl.Main
namespace OAI
/-! Extensions. -/
section
noncomputable section
open scoped TensorProduct
namespace UniversalTensorSquare
universe u
theorem irreducibles_of_kronecker_pos {n : ℕ} (lam : YoungDiagram) (hlam : lam.card = n)
(hpos : ∀ (ν : YoungDiagram) (hν : ν.card = n),
0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam)
(Saxl.canonicalTableau lam hlam) (Saxl.canonicalTableau ν hν)) :
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 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
refine ⟨Saxl.specht_irreducible _, ?_⟩
intro V _ _ _ ρ _
obtain ⟨p, ⟨e⟩⟩ := Saxl.irreducible_equiv_specht ρ
obtain ⟨f, hf⟩ := (Saxl.kronecker_pos_iff _ _ _).mp
(hpos (Saxl.partitionDiagram p) (Saxl.partitionDiagram_card p))
let := Saxl.specht_irreducible (Saxl.canonicalTableau (Saxl.partitionDiagram p)
(Saxl.partitionDiagram_card p))
have hs := Representation.IsIrreducible.injective_or_eq_zero
(W := Saxl.Specht (Saxl.canonicalTableau lam hlam) ⊗[ℂ]
Saxl.Specht (Saxl.canonicalTableau lam hlam)) f
have hi : Function.Injective f := hs.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),
(∀ (ν : YoungDiagram) (hν : ν.card = m * (m + 1) / 2),
0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam)
(Saxl.canonicalTableau lam hlam) (Saxl.canonicalTableau ν hν)) ∧
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
refine ⟨Saxl.staircase m, Saxl.staircase_card m, Saxl.saxl_kronecker_pos m hm, ?_⟩
exact irreducibles_of_kronecker_pos _ _ (Saxl.saxl_kronecker_pos m hm)
def staircaseIndex (n : ℕ) : ℕ :=
Nat.findGreatest (fun M => (Saxl.staircase M).card ≤ n ∧
(Saxl.staircase M).card % 2 = n % 2) n
lemma index_le_staircase_card (M : ℕ) : M ≤ (Saxl.staircase M).card := by
cases M with
| zero => simp
| succ M => rw [Saxl.staircase_card_succ]; omega
lemma staircaseIndex_spec (n : ℕ) :
(Saxl.staircase (staircaseIndex n)).card ≤ n ∧
(Saxl.staircase (staircaseIndex n)).card % 2 = n % 2 := by
unfold staircaseIndex
rcases Nat.mod_two_eq_zero_or_one n with he | ho
· apply Nat.findGreatest_spec (P := fun M => (Saxl.staircase M).card ≤ n ∧
(Saxl.staircase M).card % 2 = n % 2) (m := 0) (Nat.zero_le n)
simp [he]
· apply Nat.findGreatest_spec (P := fun M => (Saxl.staircase M).card ≤ n ∧
(Saxl.staircase M).card % 2 = n % 2) (m := 1) (n := n) (by omega)
simp only [Saxl.staircase_card]
norm_num
omega
lemma le_staircaseIndex (n M : ℕ) (hle : (Saxl.staircase M).card ≤ n)
(hpar : (Saxl.staircase M).card % 2 = n % 2) : M ≤ staircaseIndex n := by
apply Nat.le_findGreatest ((index_le_staircase_card M).trans hle)
exact ⟨hle, hpar⟩
def remainderPairs (n : ℕ) : ℕ := (n - (Saxl.staircase (staircaseIndex n)).card) / 2
lemma degree_eq (n : ℕ) :
(Saxl.staircase (staircaseIndex n)).card + 2 * remainderPairs n = n := by
have h := staircaseIndex_spec n
unfold remainderPairs
omega
lemma remainderPairs_bound_odd (n : ℕ) (hodd : staircaseIndex n % 2 = 1) :
2 * remainderPairs n ≤ staircaseIndex n - 1 := by
have hs := staircaseIndex_spec n
have hd := degree_eq n
have hpar : (Saxl.staircase (staircaseIndex n + 1)).card % 2 = n % 2 := by
rw [Saxl.staircase_card_succ]
omega
have hlt : n < (Saxl.staircase (staircaseIndex n + 1)).card := by
by_contra hh
have := le_staircaseIndex n (staircaseIndex n + 1) (by omega) hpar
omega
rw [Saxl.staircase_card_succ] at hlt
omega
lemma remainderPairs_bound_even (n : ℕ) (heven : staircaseIndex n % 2 = 0) :
2 * remainderPairs n ≤ 3 * staircaseIndex n + 4 := by
have hs := staircaseIndex_spec n
have hd := degree_eq n
have hsucc : (Saxl.staircase (staircaseIndex n + 3)).card =
(Saxl.staircase (staircaseIndex n)).card + 3 * staircaseIndex n + 6 := by
have h1 := Saxl.staircase_card_succ (staircaseIndex n)
have h2 : (Saxl.staircase (staircaseIndex n + 2)).card =
(Saxl.staircase (staircaseIndex n + 1)).card + (staircaseIndex n + 2) := by
simpa [Nat.add_assoc] using Saxl.staircase_card_succ (staircaseIndex n + 1)
have h3 : (Saxl.staircase (staircaseIndex n + 3)).card =
(Saxl.staircase (staircaseIndex n + 2)).card + (staircaseIndex n + 3) := by
simpa [Nat.add_assoc] using Saxl.staircase_card_succ (staircaseIndex n + 2)
omega
have hpar : (Saxl.staircase (staircaseIndex n + 3)).card % 2 = n % 2 := by
rw [hsucc]
omega
have hlt : n < (Saxl.staircase (staircaseIndex n + 3)).card := by
by_contra hh
have := le_staircaseIndex n (staircaseIndex n + 3) (by omega) hpar
omega
rw [hsucc] at hlt
omega
lemma degree_le_64_of_small_index (n : ℕ) (hM : staircaseIndex n < 9) : n ≤ 64 := by
by_contra hn
rcases Nat.mod_two_eq_zero_or_one n with he | ho
· have h11 : 11 ≤ staircaseIndex n := by
apply le_staircaseIndex
· norm_num [Saxl.staircase_card]
omega
· norm_num [Saxl.staircase_card, he]
omega
· have h9 : 9 ≤ staircaseIndex n := by
apply le_staircaseIndex
· norm_num [Saxl.staircase_card]
omega
· norm_num [Saxl.staircase_card, ho]
omega
end UniversalTensorSquare
namespace Saxl.Path
theorem odd_path_support (r : ℕ) (μ : YoungDiagram)
(t : Tableau (2*r+1) μ) (hμ : μ.colLen 0 ≤ 4) :
∃ f : Representation.IntertwiningMap (spechtRep t)
(cyclic (wordRep (2*r+1) 4) (bandWord r)).toRepresentation, f ≠ 0 := by
have hn : μ.transpose.rowLens.sum = 2*r+1 := by
simpa only [Fintype.card_fin] using
Fintype.card_congr ((Columns.columnTableau μ).trans t.symm)
obtain ⟨s, hs⟩ := fourRow_tableau_noncancellation (2*r+1) μ hn hμ
let L : Fin (μ.colLen 0) → Fin 4 → ℂ := fun i a =>
Band.conjugation (Band.basis i.val) (parts a).1 (parts a).2
apply hom_to_cyclic_of_pair L (spechtSub t)
⟨polytabloid s, polytabloid_mem_all t s⟩ (bandWord r) (bandWord_real r)
intro h
rw [chainFunctional_endpoint] at hs
exact hs (by change (-1 : ℂ)^r * dotProduct (wordMap L (polytabloid s)) (bandWord r) = 0
rw [h, mul_zero])
end Saxl.Path
end
end
end OAI
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