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c8720e4 | 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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import Mathlib
import OAI.RepresentationTheory.PartialPermutation.Model
import OAI.RepresentationTheory.PartialPermutation.ModuleTransport
namespace OAI
section
open scoped Classical
open scoped BigOperators ComplexConjugate MonoidAlgebra
open scoped BigOperators ComplexConjugate
open scoped MonoidAlgebra BigOperators
namespace PartialPermutation
noncomputable section
section Components
variable {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
[FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
omit [Fintype G] [FiniteDimensional ℂ V] in
lemma componentSimple_simple (c : isotypicComponents ℂ[G] ρ.asModule) :
IsSimpleModule ℂ[G] (componentSimple ρ c) := c.property.choose_spec.1
omit [Fintype G] [FiniteDimensional ℂ V] in
lemma componentSimple_component (c : isotypicComponents ℂ[G] ρ.asModule) :
c.val = isotypicComponent ℂ[G] ρ.asModule (componentSimple ρ c) :=
c.property.choose_spec.2
noncomputable def componentCharacter (c : isotypicComponents ℂ[G] ρ.asModule) : OccurringTypes ρ :=
⟨(Subrepresentation.ofSubmodule' (componentSimple ρ c)).toRepresentation.character,
Subrepresentation.ofSubmodule' (componentSimple ρ c),
(subrep_irreducible_iff ρ _).mpr (componentSimple_simple ρ c), rfl⟩
lemma componentCharacter_injective : Function.Injective (componentCharacter ρ) := by
intro c d h
let σ := Subrepresentation.ofSubmodule' (componentSimple ρ c)
let τ := Subrepresentation.ofSubmodule' (componentSimple ρ d)
have : Representation.IsIrreducible σ.toRepresentation :=
(subrep_irreducible_iff ρ σ).mpr (componentSimple_simple ρ c)
have : Representation.IsIrreducible τ.toRepresentation :=
(subrep_irreducible_iff ρ τ).mpr (componentSimple_simple ρ d)
obtain ⟨e⟩ := irreducible_equiv_of_character_eq σ.toRepresentation τ.toRepresentation
(congrArg Subtype.val h)
apply Subtype.ext
rw [componentSimple_component ρ c, componentSimple_component ρ d]
exact (moduleEquivOfSubrepEquiv ρ τ σ e).symm.isotypicComponent_eq
omit [Fintype G] in
lemma componentCharacter_surjective : Function.Surjective (componentCharacter ρ) := by
intro χ
obtain ⟨σ, hirr, hχ⟩ := χ.property
have := hirr
have : IsSimpleModule ℂ[G] σ.asSubmodule := (subrep_irreducible_iff ρ σ).mp hirr
let c : isotypicComponents ℂ[G] ρ.asModule :=
⟨isotypicComponent ℂ[G] ρ.asModule σ.asSubmodule, σ.asSubmodule, inferInstance, rfl⟩
have := componentSimple_simple ρ c
have hs : σ.asSubmodule ≤ c.val := σ.asSubmodule.le_isotypicComponent
rw [componentSimple_component ρ c] at hs
obtain ⟨e⟩ := isIsotypicOfType_submodule_iff.mp
(IsIsotypicOfType.isotypicComponent ℂ[G] ρ.asModule (componentSimple ρ c))
σ.asSubmodule hs
refine ⟨c, Subtype.ext ?_⟩
change (Subrepresentation.ofSubmodule' (componentSimple ρ c)).toRepresentation.character = χ.val
rw [hχ]
exact Representation.char_iso
(subrepEquivOfModuleEquiv ρ (Subrepresentation.ofSubmodule' (componentSimple ρ c)) σ e.symm)
noncomputable def componentTypeEquiv : isotypicComponents ℂ[G] ρ.asModule ≃ OccurringTypes ρ :=
Equiv.ofBijective (componentCharacter ρ)
⟨componentCharacter_injective ρ, componentCharacter_surjective ρ⟩
instance occurringTypes_finite : Finite (OccurringTypes ρ) :=
Finite.of_surjective (componentCharacter ρ) (componentCharacter_surjective ρ)
lemma card_components_eq_types :
Nat.card (isotypicComponents ℂ[G] ρ.asModule) = Nat.card (OccurringTypes ρ) :=
Nat.card_congr (componentTypeEquiv ρ)
end Components
end
end PartialPermutation
end
end OAI
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