| import Mathlib |
| import OAI.RepresentationTheory.Saxl.Main |
| import OAI.RepresentationTheory.UniversalSquare.Finite.DegreeFive |
| import OAI.RepresentationTheory.UniversalSquare.Finite.DegreeSeven |
| import OAI.RepresentationTheory.UniversalSquare.Finite.DegreeEight |
|
|
| namespace OAI |
|
|
| /-! Small Degrees. -/ |
|
|
| section |
|
|
| namespace UniversalTensorSquare |
| open Saxl |
|
|
| universe u |
|
|
| theorem universal_tensor_square_le_ten |
| (n : ℕ) (hn : 0 < n) (h2 : n ≠ 2) (h4 : n ≠ 4) (h9 : n ≠ 9) |
| (hsmall : n ≤ 10) : |
| ∃ (lam : YoungDiagram) (hlam : lam.card = n), |
| (∀ (ν : YoungDiagram) (hν : ν.card = n), |
| 0 < kronecker (canonicalTableau lam hlam) |
| (canonicalTableau lam hlam) (canonicalTableau ν hν)) ∧ |
| Representation.IsIrreducible (spechtRep (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 |
| ((spechtRep (canonicalTableau lam hlam)).tprod |
| (spechtRep (canonicalTableau lam hlam))), Function.Injective F := by |
| have hcases : n = 1 ∨ n = 3 ∨ n = 5 ∨ n = 6 ∨ n = 7 ∨ n = 8 ∨ n = 10 := by omega |
| rcases hcases with rfl | rfl | rfl | rfl | rfl | rfl | rfl |
| · exact universal_tensor_square_triangular 1 (by decide) |
| · exact universal_tensor_square_triangular 2 (by decide) |
| · exact universal_tensor_square_five |
| · exact universal_tensor_square_triangular 3 (by decide) |
| · exact universal_tensor_square_seven |
| · exact universal_tensor_square_eight |
| · exact universal_tensor_square_triangular 4 (by decide) |
|
|
| end UniversalTensorSquare |
| end |
|
|
| end OAI |
|
|
