| import Mathlib |
| import OAI.Probability.BinarySweep.Representations.IrrepTransport |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps.Irrep |
| variable {G I : Type*} [Group G] [Fintype G] [Fintype I] |
| {V : I → Type*} [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)] |
| [∀ i, FiniteDimensional ℂ (V i)] |
| (σ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (σ i)] |
|
|
| theorem linearIndependent_classCharacters |
| (hdiff : ∀ i j, Nonempty (Representation.Equiv (σ i) (σ j)) → i = j) : |
| LinearIndependent ℂ (fun i => classCharacter (σ i)) := by |
| apply Fintype.linearIndependent_iff.mpr |
| intro a ha i |
| have h := congrArg (classPairRight (classCharacter (σ i))) ha |
| simp only [map_sum, map_smul, map_zero, smul_eq_mul] at h |
| have hi (j : I) : Nonempty (Representation.Equiv (σ i) (σ j)) ↔ i = j := by |
| constructor |
| · exact hdiff i j |
| · rintro rfl |
| exact ⟨Representation.Equiv.refl _⟩ |
| simp_rw [classPair_characters, hi] at h |
| simpa only [mul_ite, mul_one, mul_zero, Finset.sum_ite_eq, Finset.mem_univ, ite_true] using h |
|
|
| theorem complete_of_card_ge [Nonempty I] |
| (hdiff : ∀ i j, Nonempty (Representation.Equiv (σ i) (σ j)) → i = j) |
| (hcard : Nat.card (ConjClasses G) ≤ Fintype.card I) |
| {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W] |
| (ρ : Representation ℂ G W) [ρ.IsIrreducible] : |
| ∃ i, Nonempty (Representation.Equiv ρ (σ i)) := by |
| let : Fintype (ConjClasses G) := Fintype.ofFinite _ |
| have hli := linearIndependent_classCharacters σ hdiff |
| have hdim : Fintype.card I = Module.finrank ℂ (ConjClasses G → ℂ) := by |
| apply le_antisymm hli.fintype_card_le_finrank |
| simpa only [Module.finrank_pi, Nat.card_eq_fintype_card] using hcard |
| have ht := hli.span_eq_top_of_card_eq_finrank hdim |
| by_contra hn |
| push Not at hn |
| have hzero (i : I) : classPairRight (classCharacter ρ) (classCharacter (σ i)) = 0 := by |
| have hi : ¬ Nonempty (Representation.Equiv ρ (σ i)) := fun ⟨f⟩ => (hn i).false f |
| rw [classPair_characters, ite_eq_right hi] |
| have hle : Submodule.span ℂ (Set.range (fun i => classCharacter (σ i))) ≤ |
| (classPairRight (classCharacter ρ)).ker := by |
| rw [Submodule.span_le] |
| rintro _ ⟨i, rfl⟩ |
| exact hzero i |
| rw [ht] at hle |
| have he : classPairRight (classCharacter ρ) (classCharacter ρ) = 0 := |
| hle (Submodule.mem_top) |
| rw [classPair_character_self] at he |
| exact one_ne_zero he |
|
|
| end BinaryCoordinateSweeps.Irrep |
|
|
| end |
|
|
| end OAI |
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|
