| import Mathlib |
| import OAI.Probability.BinarySweep.YoungTheory.YoungFinite |
| import OAI.Probability.BinarySweep.Representations.CharacterComplete |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace BinaryCoordinateSweeps.Young |
|
|
| theorem partitionRep_complete {n : ℕ} {V : Type*} [AddCommGroup V] [Module ℂ V] |
| [FiniteDimensional ℂ V] (ρ : Representation ℂ (Equiv.Perm (Fin n)) V) [ρ.IsIrreducible] : |
| ∃ p : n.Partition, Nonempty (Representation.Equiv ρ (partitionRep p)) := by |
| refine Irrep.complete_of_card_ge (G := Equiv.Perm (Fin n)) |
| (V := fun p : n.Partition => PartitionSpace p) |
| (fun p : n.Partition => partitionRep p) ?_ ?_ ρ |
| · intro p q ⟨f⟩ |
| exact partitionRep_equiv_injective p q f |
| · let : Fintype (ConjClasses (Equiv.Perm (Fin n))) := Fintype.ofFinite _ |
| have h := Fintype.card_le_of_injective (conjPartition (X := Fin n)) conjPartition_injective |
| simpa only [Nat.card_eq_fintype_card, Fintype.card_fin] using h |
|
|
| theorem specht_complete {X V : Type*} [Fintype X] [DecidableEq X] |
| [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] |
| (ρ : Representation ℂ (Equiv.Perm X) V) [ρ.IsIrreducible] : |
| ∃ (μ : YoungDiagram) (e : Cell μ ≃ X), Nonempty (Representation.Equiv ρ |
| ((spechtRep μ).comp e.symm.permCongrHom.toMonoidHom)) := by |
| let a : X ≃ Fin (Fintype.card X) := Fintype.equivFin X |
| let ρ' : Representation ℂ (Equiv.Perm (Fin (Fintype.card X))) V := |
| ρ.comp a.symm.permCongrHom.toMonoidHom |
| have : ρ'.IsIrreducible := Irrep.irreducible_comp_equiv _ _ |
| obtain ⟨p, ⟨f⟩⟩ := partitionRep_complete ρ' |
| let e : Cell (diagram p) ≃ X := (cellsEquivFin p).trans a.symm |
| refine ⟨diagram p, e, ⟨Representation.Equiv.mk f.toLinearEquiv ?_⟩⟩ |
| intro g |
| apply LinearMap.ext |
| intro v |
| have h := Representation.IntertwiningMap.isIntertwining ρ' (partitionRep p) |
| f.toIntertwiningMap (a.permCongr g) v |
| change f.toLinearEquiv (ρ (a.symm.permCongr (a.permCongr g)) v) = |
| spechtRep (diagram p) ((cellsEquivFin p).symm.permCongr (a.permCongr g)) (f.toLinearEquiv v) at h |
| rw [show a.symm.permCongr (a.permCongr g) = g from |
| a.permCongr.symm_apply_apply g] at h |
| exact h |
|
|
| end BinaryCoordinateSweeps.Young |
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|
| end |
|
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| end OAI |
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|
