| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators MonoidAlgebra | |
| namespace BinaryCoordinateSweeps | |
| namespace Irrep | |
| variable {G V : Type*} [Group G] [Fintype G] | |
| [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] | |
| (ρ : Representation ℂ G V) [ρ.IsIrreducible] | |
| omit [Fintype G] [FiniteDimensional ℂ V] in | |
| lemma invariant_eq_zero_of_finrank_gt_one (hD : 1 < Module.finrank ℂ V) | |
| (v : V) (hv : ∀ g, ρ g v = v) : v = 0 := by | |
| let f : Representation.IntertwiningMap (1 : Representation ℂ G ℂ) ρ := | |
| (LinearMap.toSpanSingleton ℂ V v).intertwiningMap_of_isIntertwiningMap _ _ (by | |
| intro g c | |
| simp only [MonoidHom.one_apply, Module.End.one_apply, LinearMap.toSpanSingleton_apply, | |
| map_smul, hv]) | |
| rcases Representation.IsIrreducible.surjective_or_eq_zero f with hf | hf | |
| · have hd := LinearMap.finrank_le_finrank_of_surjective (f := f.toLinearMap) hf | |
| simp only [Module.finrank_self] at hd | |
| omega | |
| · have hh := congrArg (fun a : Representation.IntertwiningMap | |
| (1 : Representation ℂ G ℂ) ρ => a (1 : ℂ)) hf | |
| simpa [f] using hh | |
| omit [FiniteDimensional ℂ V] in | |
| lemma sum_apply_eq_zero_of_finrank_gt_one (hD : 1 < Module.finrank ℂ V) (x : V) : | |
| ∑ g : G, ρ g x = 0 := by | |
| apply invariant_eq_zero_of_finrank_gt_one ρ hD | |
| intro g | |
| simp only [map_sum, ← Module.End.mul_apply, ← map_mul] | |
| exact Function.Bijective.sum_comp (Group.mulLeft_bijective g) (fun h : G => ρ h x) | |
| omit [FiniteDimensional ℂ V] in | |
| lemma sum_eq_zero_of_finrank_gt_one (hD : 1 < Module.finrank ℂ V) : | |
| ∑ g : G, ρ g = 0 := by | |
| ext x | |
| simpa using sum_apply_eq_zero_of_finrank_gt_one ρ hD x | |
| include ρ in | |
| omit [FiniteDimensional ℂ V] in | |
| lemma finrank_le_card : Module.finrank ℂ V ≤ Fintype.card G := by | |
| let : IsSimpleModule ℂ[G] ρ.asModule := inferInstance | |
| let : Nontrivial ρ.asModule := IsSimpleModule.nontrivial ℂ[G] ρ.asModule | |
| obtain ⟨v, hv⟩ := exists_ne (0 : ρ.asModule) | |
| let f : ℂ[G] →ₗ[ℂ] ρ.asModule := | |
| (LinearMap.toSpanSingleton ℂ[G] ρ.asModule v).restrictScalars ℂ | |
| have hf : Function.Surjective f := IsSimpleModule.toSpanSingleton_surjective ℂ[G] hv | |
| have hd := LinearMap.finrank_le_finrank_of_surjective (f := f) hf | |
| have he : Module.finrank ℂ ρ.asModule = Module.finrank ℂ V := | |
| ρ.asModuleEquiv.finrank_eq | |
| rw [he] at hd | |
| rw [Module.finrank_eq_card_basis (MonoidAlgebra.basis G ℂ)] at hd | |
| exact hd | |
| end Irrep | |
| end BinaryCoordinateSweeps | |
| end | |
| end OAI | |