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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 87 88 89 90 91 | import Mathlib
namespace OAI
section
open scoped Classical
open scoped BigOperators ComplexConjugate MonoidAlgebra
open scoped BigOperators ComplexConjugate
open scoped MonoidAlgebra BigOperators
namespace PartialPermutation
noncomputable section
section ModuleTransport
variable {G V : Type*} [Group G] [AddCommGroup V] [Module ℂ V]
(ρ : Representation ℂ G V)
noncomputable def subrepModuleEquiv (σ : Subrepresentation ρ) :
σ.toRepresentation.asModule ≃ₗ[ℂ[G]] σ.asSubmodule where
toFun x := ⟨x.1, x.2⟩
invFun x := ⟨x.1, x.2⟩
left_inv _ := rfl
right_inv _ := rfl
map_add' _ _ := rfl
map_smul' a x := by
apply Subtype.ext
change ((σ.toRepresentation.asAlgebraHom a) x).val =
(ρ.asAlgebraHom a) x.val
induction a using MonoidAlgebra.induction_linear with
| zero => simp only [map_zero]; rfl
| add a b ha hb =>
simp only [map_add]
change ((σ.toRepresentation.asAlgebraHom a) x).val +
((σ.toRepresentation.asAlgebraHom b) x).val = _ + _
rw [ha, hb]
| single g c =>
rw [Representation.asAlgebraHom_single, Representation.asAlgebraHom_single]
rfl
lemma subrep_irreducible_iff (σ : Subrepresentation ρ) :
Representation.IsIrreducible σ.toRepresentation ↔ IsSimpleModule ℂ[G] σ.asSubmodule := by
rw [Representation.irreducible_iff_isSimpleModule_asModule]
exact ⟨fun _ => IsSimpleModule.congr (subrepModuleEquiv ρ σ).symm,
fun _ => IsSimpleModule.congr (subrepModuleEquiv ρ σ)⟩
noncomputable def subrepEquivOfModuleEquiv (σ τ : Subrepresentation ρ)
(e : σ.asSubmodule ≃ₗ[ℂ[G]] τ.asSubmodule) :
Representation.Equiv σ.toRepresentation τ.toRepresentation := by
let e' := (subrepModuleEquiv ρ σ).trans (e.trans (subrepModuleEquiv ρ τ).symm)
exact ((Representation.IntertwiningMap.equivLinearMapAsModule _ _).symm
e'.toLinearMap).ofBijective e'.bijective
noncomputable def moduleEquivOfSubrepEquiv (σ τ : Subrepresentation ρ)
(e : Representation.Equiv σ.toRepresentation τ.toRepresentation) :
σ.asSubmodule ≃ₗ[ℂ[G]] τ.asSubmodule := by
let f := Representation.IntertwiningMap.equivLinearMapAsModule _ _ e.toIntertwiningMap
let e' := LinearEquiv.ofBijective f e.bijective
exact (subrepModuleEquiv ρ σ).symm.trans (e'.trans (subrepModuleEquiv ρ τ))
end ModuleTransport
lemma irreducible_equiv_of_character_eq {G V W : Type*} [Group G] [Fintype G]
[AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
[AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
(ρ : Representation ℂ G V) (σ : Representation ℂ G W)
[Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
(h : ρ.character = σ.character) : Nonempty (Representation.Equiv σ ρ) := by
classical
let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
exact_mod_cast (Nat.card_pos (α := G)).ne')
have hself := Representation.char_orthonormal ρ ρ
have hother := Representation.char_orthonormal ρ σ
rw [h] at hself
rw [h] at hother
have he : Nonempty (ρ.Equiv ρ) := ⟨Representation.Equiv.refl ρ⟩
simp only [he, ite_eq_left] at hself
by_contra hn
simp only [hn] at hother
exact one_ne_zero (hself.symm.trans hother)
def OccurringTypes {G V : Type*} [Group G] [AddCommGroup V] [Module ℂ V]
[FiniteDimensional ℂ V] (ρ : Representation ℂ G V) :=
{χ : G → ℂ // ∃ σ : Subrepresentation ρ,
Representation.IsIrreducible σ.toRepresentation ∧ χ = σ.toRepresentation.character}
end
end PartialPermutation
end
end OAI
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