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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | 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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