import Mathlib import OAI.RepresentationTheory.Saxl.Main import OAI.RepresentationTheory.UniversalSquare.Support.Asymptotic import OAI.RepresentationTheory.UniversalSquare.Support.ExceptionalFourteen import OAI.RepresentationTheory.UniversalSquare.Support.SmallDegrees import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree11 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree12 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree13 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree14 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree16 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree17 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree18 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree19 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree20 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree22 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree23 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree24 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree25 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree26 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree27 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree29 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree30 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree31 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree32 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree33 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree34 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree35 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree37 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree38 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree39 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree40 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree41 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree42 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree43 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree44 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree46 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree47 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree48 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree49 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree50 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree51 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree52 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree53 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree54 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree56 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree57 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree58 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree59 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree60 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree61 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree62 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree63 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree64 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree89 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree112 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree116 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree118 import OAI.RepresentationTheory.UniversalSquare.Finite.NumericalDegree151 namespace OAI /-! Universal tensor squares. -/ section noncomputable section namespace UniversalTensorSquare open Saxl lemma universal_partition_le64 (n : ℕ) (hn : 0 < n) (h2 : n ≠ 2) (h4 : n ≠ 4) (h9 : n ≠ 9) (hsmall : n ≤ 64) : ∃ (lam : YoungDiagram) (hc : lam.card = n), ∀ (μ : YoungDiagram) (hμ : μ.card = n), 0 < kronecker (canonicalTableau lam hc) (canonicalTableau lam hc) (canonicalTableau μ hμ) := by by_cases hten : n ≤ 10 · obtain ⟨lam,hc,hpos,_⟩ := universal_tensor_square_le_ten.{0} n hn h2 h4 h9 hten exact ⟨lam,hc,hpos⟩ have hlo : 11 ≤ n := by omega interval_cases n · exact ⟨_,degreeCard11,degree_pos11⟩ · exact ⟨_,degreeCard12,degree_pos12⟩ · exact ⟨_,degreeCard13,degree_pos13⟩ · exact ⟨_,degreeCard14,degree_pos14⟩ · exact ⟨staircase 5,staircase_card 5,saxl_kronecker_pos 5 (by decide)⟩ · exact ⟨_,degreeCard16,degree_pos16⟩ · exact ⟨_,degreeCard17,degree_pos17⟩ · exact ⟨_,degreeCard18,degree_pos18⟩ · exact ⟨_,degreeCard19,degree_pos19⟩ · exact ⟨_,degreeCard20,degree_pos20⟩ · exact ⟨staircase 6,staircase_card 6,saxl_kronecker_pos 6 (by decide)⟩ · exact ⟨_,degreeCard22,degree_pos22⟩ · exact ⟨_,degreeCard23,degree_pos23⟩ · exact ⟨_,degreeCard24,degree_pos24⟩ · exact ⟨_,degreeCard25,degree_pos25⟩ · exact ⟨_,degreeCard26,degree_pos26⟩ · exact ⟨_,degreeCard27,degree_pos27⟩ · exact ⟨staircase 7,staircase_card 7,saxl_kronecker_pos 7 (by decide)⟩ · exact ⟨_,degreeCard29,degree_pos29⟩ · exact ⟨_,degreeCard30,degree_pos30⟩ · exact ⟨_,degreeCard31,degree_pos31⟩ · exact ⟨_,degreeCard32,degree_pos32⟩ · exact ⟨_,degreeCard33,degree_pos33⟩ · exact ⟨_,degreeCard34,degree_pos34⟩ · exact ⟨_,degreeCard35,degree_pos35⟩ · exact ⟨staircase 8,staircase_card 8,saxl_kronecker_pos 8 (by decide)⟩ · exact ⟨_,degreeCard37,degree_pos37⟩ · exact ⟨_,degreeCard38,degree_pos38⟩ · exact ⟨_,degreeCard39,degree_pos39⟩ · exact ⟨_,degreeCard40,degree_pos40⟩ · exact ⟨_,degreeCard41,degree_pos41⟩ · exact ⟨_,degreeCard42,degree_pos42⟩ · exact ⟨_,degreeCard43,degree_pos43⟩ · exact ⟨_,degreeCard44,degree_pos44⟩ · exact ⟨staircase 9,staircase_card 9,saxl_kronecker_pos 9 (by decide)⟩ · exact ⟨_,degreeCard46,degree_pos46⟩ · exact ⟨_,degreeCard47,degree_pos47⟩ · exact ⟨_,degreeCard48,degree_pos48⟩ · exact ⟨_,degreeCard49,degree_pos49⟩ · exact ⟨_,degreeCard50,degree_pos50⟩ · exact ⟨_,degreeCard51,degree_pos51⟩ · exact ⟨_,degreeCard52,degree_pos52⟩ · exact ⟨_,degreeCard53,degree_pos53⟩ · exact ⟨_,degreeCard54,degree_pos54⟩ · exact ⟨staircase 10,staircase_card 10,saxl_kronecker_pos 10 (by decide)⟩ · exact ⟨_,degreeCard56,degree_pos56⟩ · exact ⟨_,degreeCard57,degree_pos57⟩ · exact ⟨_,degreeCard58,degree_pos58⟩ · exact ⟨_,degreeCard59,degree_pos59⟩ · exact ⟨_,degreeCard60,degree_pos60⟩ · exact ⟨_,degreeCard61,degree_pos61⟩ · exact ⟨_,degreeCard62,degree_pos62⟩ · exact ⟨_,degreeCard63,degree_pos63⟩ · exact ⟨_,degreeCard64,degree_pos64⟩ lemma universal_partition_exceptional (n : ℕ) (hexc : exceptionalPair (staircaseIndex n) (remainderPairs n)) : ∃ (lam : YoungDiagram) (hc : lam.card = n), ∀ (μ : YoungDiagram) (hμ : μ.card = n), 0 < kronecker (canonicalTableau lam hc) (canonicalTableau lam hc) (canonicalTableau μ hμ) := by have he := degree_eq n rw [staircase_card] at he have hcases : n = 71 ∨ n = 73 ∨ n = 75 ∨ n = 77 ∨ n = 79 ∨ n = 81 ∨ n = 83 ∨ n = 85 ∨ n = 87 ∨ n = 89 ∨ n = 106 ∨ n = 108 ∨ n = 110 ∨ n = 112 ∨ n = 114 ∨ n = 116 ∨ n = 118 ∨ n = 149 ∨ n = 151 := by unfold exceptionalPair at hexc rcases hexc with ⟨hM,hr⟩ | ⟨hM,hr⟩ | ⟨hM,hr⟩ <;> rw [hM] at he <;> norm_num at he <;> omega rcases hcases with rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl · exact ⟨_,exceptionalCard71,exceptional_pos71⟩ · exact ⟨_,exceptionalCard73,exceptional_pos73⟩ · exact ⟨_,exceptionalCard75,exceptional_pos75⟩ · exact ⟨_,exceptionalCard77,exceptional_pos77⟩ · exact ⟨_,exceptionalCard79,exceptional_pos79⟩ · exact ⟨_,exceptionalCard81,exceptional_pos81⟩ · exact ⟨_,exceptionalCard83,exceptional_pos83⟩ · exact ⟨_,exceptionalCard85,exceptional_pos85⟩ · exact ⟨_,exceptionalCard87,exceptional_pos87⟩ · exact ⟨_,degreeCard89,degree_pos89⟩ · exact ⟨_,exceptionalCard106,exceptional_pos106⟩ · exact ⟨_,exceptionalCard108,exceptional_pos108⟩ · exact ⟨_,exceptionalCard110,exceptional_pos110⟩ · exact ⟨_,degreeCard112,degree_pos112⟩ · exact ⟨_,exceptionalCard114,exceptional_pos114⟩ · exact ⟨_,degreeCard116,degree_pos116⟩ · exact ⟨_,degreeCard118,degree_pos118⟩ · exact ⟨_,exceptionalCard149,exceptional_pos149⟩ · exact ⟨_,degreeCard151,degree_pos151⟩ universe u theorem universal_tensor_square (n : ℕ) (hn : 0 < n) (h2 : n ≠ 2) (h4 : n ≠ 4) (h9 : n ≠ 9) : ∃ (lam : YoungDiagram) (hlam : lam.card = n), (∀ (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 have hex : ∃ (lam : YoungDiagram) (hc : lam.card = n), ∀ (μ : YoungDiagram) (hμ : μ.card = n), 0 < kronecker (canonicalTableau lam hc) (canonicalTableau lam hc) (canonicalTableau μ hμ) := by by_cases hsmall : n ≤ 64 · exact universal_partition_le64 n hn h2 h4 h9 hsmall · have hlarge : 64 < n := by omega by_cases hexc : exceptionalPair (staircaseIndex n) (remainderPairs n) · exact universal_partition_exceptional n hexc · have hM := (fixedCandidate_large_degree n hlarge).1 exact ⟨fixedCandidate n,fixedCandidate_card n (by omega), fixedCandidate_nonexceptional_pos n hlarge hexc⟩ obtain ⟨lam,hc,hpos⟩ := hex exact ⟨lam,hc,hpos,irreducibles_of_kronecker_pos lam hc hpos⟩ end UniversalTensorSquare end end namespace UniversalTensorSquares universe u theorem universal_tensor_square (n : ℕ) (hn : 0 < n) (h2 : n ≠ 2) (h4 : n ≠ 4) (h9 : n ≠ 9) : ∃ (lam : YoungDiagram) (hlam : lam.card = n), (∀ (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 exact UniversalTensorSquare.universal_tensor_square n hn h2 h4 h9 end UniversalTensorSquares end OAI