| import Mathlib |
| import OAI.RepresentationTheory.Saxl.Main |
|
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| namespace OAI |
|
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| /-! Extensions. -/ |
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| section |
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| noncomputable section |
| open scoped TensorProduct |
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| namespace UniversalTensorSquare |
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| universe u |
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|
| 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 |
|
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| 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 |
|
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| end OAI |
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