| 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 |
|
|
