fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
indToCoindAux_of_not_rel (g g₁ : G) (a : A) (h : ¬(QuotientGroup.rightRel S).r g₁ g) :
indToCoindAux A g a g₁ = 0 := by
simp [indToCoindAux, dif_neg h]
@[simp] | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | indToCoindAux_of_not_rel | null |
indToCoindAux_mul_snd (g g₁ : G) (a : A) (s : S) :
indToCoindAux A g a (s * g₁) = A.ρ s (indToCoindAux A g a g₁) := by
rcases em ((QuotientGroup.rightRel S).r g₁ g) with ⟨s₁, rfl⟩ | h
· simp only [indToCoindAux, LinearMap.pi_apply]
rw [dif_pos ⟨s * s₁, mul_assoc ..⟩, dif_pos ⟨s₁, rfl⟩]
simp [S.1.smul_de... | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | indToCoindAux_mul_snd | null |
indToCoindAux_mul_fst (g₁ g₂ : G) (a : A) (s : S) :
indToCoindAux A (s * g₁) (A.ρ s a) g₂ = indToCoindAux A g₁ a g₂ := by
rcases em ((QuotientGroup.rightRel S).r g₂ g₁) with ⟨s₁, rfl⟩ | h
· simp only [indToCoindAux, LinearMap.pi_apply]
rw [dif_pos ⟨s₁ * s⁻¹, by simp [S.1.smul_def, smul_eq_mul, mul_assoc]⟩,... | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | indToCoindAux_mul_fst | null |
indToCoindAux_snd_mul_inv (g₁ g₂ g₃ : G) (a : A) :
indToCoindAux A g₁ a (g₂ * g₃⁻¹) = indToCoindAux A (g₁ * g₃) a g₂ := by
rcases em ((QuotientGroup.rightRel S).r (g₂ * g₃⁻¹) g₁) with ⟨s, hs⟩ | h
· simp [S.1.smul_def, mul_assoc, ← eq_mul_inv_iff_mul_eq.1 hs]
· rw [indToCoindAux_of_not_rel (h := h), indToCoind... | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | indToCoindAux_snd_mul_inv | null |
indToCoindAux_fst_mul_inv (g₁ g₂ g₃ : G) (a : A) :
indToCoindAux A (g₁ * g₂⁻¹) a g₃ = indToCoindAux A g₁ a (g₃ * g₂) := by
simpa using (indToCoindAux_snd_mul_inv g₁ g₃ g₂⁻¹ a).symm | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | indToCoindAux_fst_mul_inv | null |
indToCoindAux_comm {A B : Rep k S} (f : A ⟶ B) (g₁ g₂ : G) (a : A) :
indToCoindAux B g₁ (f.hom a) g₂ = f.hom (indToCoindAux A g₁ a g₂) := by
rcases em ((QuotientGroup.rightRel S).r g₂ g₁) with ⟨s, rfl⟩ | h
· simp [S.1.smul_def, hom_comm_apply]
· simp [indToCoindAux_of_not_rel (h := h)]
variable (A) in | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | indToCoindAux_comm | null |
noncomputable indToCoind :
ind S.subtype A →ₗ[k] coind S.subtype A :=
Representation.Coinvariants.lift _ (TensorProduct.lift <| linearCombination _ fun g =>
LinearMap.codRestrict _ (indToCoindAux A g) fun _ _ _ => by simp) fun _ => by ext; simp
variable [S.FiniteIndex]
attribute [local instance] Subgroup.fint... | abbrev | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | indToCoind | Let `S ≤ G` be a subgroup and `A` a `k`-linear `S`-representation. This is the `k`-linear map
`Ind_S^G(A) →ₗ[k] Coind_S^G(A)` sending `(⟦g ⊗ₜ[k] a⟧, sg) ↦ ρ(s)(a)`. |
@[simps]
noncomputable coindToInd : coind S.subtype A →ₗ[k] ind S.subtype A where
toFun f := ∑ g : Quotient (QuotientGroup.rightRel S), Quotient.liftOn g (fun g =>
IndV.mk S.subtype _ g (f.1 g)) fun g₁ g₂ ⟨s, (hs : _ * _ = _)⟩ =>
(Submodule.Quotient.eq _).2 <| Coinvariants.mem_ker_of_eq s
(single g₂... | def | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | coindToInd | Let `S ≤ G` be a finite index subgroup, `g₁, ..., gₙ` a set of right coset representatives of
`S`, and `A` a `k`-linear `S`-representation. This is the `k`-linear map
`Coind_S^G(A) →ₗ[k] Ind_S^G(A)` sending `f : G → A` to `∑ᵢ ⟦gᵢ ⊗ₜ[k] f(gᵢ)⟧` for `1 ≤ i ≤ n`. |
coindToInd_of_support_subset_orbit (g : G) (f : coind S.subtype A)
(hx : f.1.support ⊆ MulAction.orbit S g) :
coindToInd A f = IndV.mk S.subtype _ g (f.1 g) := by
rw [coindToInd_apply, Finset.sum_eq_single ⟦g⟧]
· simp
· intro b _ hb
induction b using Quotient.inductionOn with | h b =>
have : f.1 b... | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | coindToInd_of_support_subset_orbit | null |
@[simps! hom_hom_hom inv_hom_hom]
noncomputable indCoindIso : ind S.subtype A ≅ coind S.subtype A :=
Action.mkIso ({
hom := ModuleCat.ofHom <| indToCoind A
inv := ModuleCat.ofHom <| coindToInd A
hom_inv_id := by
ext g a
simp only [ModuleCat.hom_comp, ModuleCat.hom_ofHom, LinearMap.coe_comp, Fu... | def | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | indCoindIso | Let `S ≤ G` be a finite index subgroup, `g₁, ..., gₙ` a set of right coset representatives of
`S`, and `A` a `k`-linear `S`-representation. This is an isomorphism `Ind_S^G(A) ≅ Coind_S^G(A)`.
The forward map sends `(⟦g ⊗ₜ[k] a⟧, sg) ↦ ρ(s)(a)`, and the inverse sends `f : G → A` to
`∑ᵢ ⟦gᵢ ⊗ₜ[k] f(gᵢ)⟧` for `1 ≤ i ≤ n`. |
@[simps! hom_app inv_app]
noncomputable indCoindNatIso : indFunctor k S.subtype ≅ coindFunctor k S.subtype :=
NatIso.ofComponents (fun _ => indCoindIso _) fun f => by
simp only [indFunctor_obj, coindFunctor_obj]; ext; simp [indToCoindAux_comm] | def | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | indCoindNatIso | Given a finite index subgroup `S ≤ G`, this is a natural isomorphism between the `Ind_S^G` and
`Coind_G^S` functors `Rep k S ⥤ Rep k G`. |
noncomputable resIndAdjunction : Action.res _ S.subtype ⊣ indFunctor k S.subtype :=
(resCoindAdjunction k S.subtype).ofNatIsoRight (indCoindNatIso k S).symm | def | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | resIndAdjunction | Given a finite index subgroup `S ≤ G`, `Ind_S^G` is right adjoint to the restriction functor
`Res k G ⥤ Res k S`, since it is naturally isomorphic to `Coind_S^G`. |
@[simp]
resIndAdjunction_counit_app :
(resIndAdjunction k S).counit.app A = (Action.res _ S.subtype).map (indCoindIso A).hom ≫
(resCoindAdjunction k S.subtype).counit.app A := rfl
@[simp] | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | resIndAdjunction_counit_app | null |
resIndAdjunction_unit_app (B : Rep k G) :
(resIndAdjunction k S).unit.app B = (resCoindAdjunction k S.subtype).unit.app B ≫
(indCoindIso ((Action.res _ S.subtype).obj B)).inv := rfl | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | resIndAdjunction_unit_app | null |
resIndAdjunction_homEquiv_apply
{B : Rep k G} (f : (Action.res _ S.subtype).obj B ⟶ A) :
(resIndAdjunction k S).homEquiv _ _ f =
resCoindHomEquiv S.subtype B A f ≫ (indCoindIso A).inv := by
simp only [resIndAdjunction, Adjunction.ofNatIsoRight, resCoindAdjunction,
Adjunction.mkOfHomEquiv_homEquiv]
... | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | resIndAdjunction_homEquiv_apply | null |
resIndAdjunction_homEquiv_symm_apply
{B : Rep k G} (f : B ⟶ (indFunctor k S.subtype).obj A) :
((resIndAdjunction k S).homEquiv _ _).symm f =
(resCoindHomEquiv S.subtype B A).symm (f ≫ (indCoindIso A).hom) := by
simp only [resIndAdjunction, Adjunction.ofNatIsoRight, resCoindAdjunction,
Adjunction.mkO... | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | resIndAdjunction_homEquiv_symm_apply | null |
noncomputable coindResAdjunction : coindFunctor k S.subtype ⊣ Action.res _ S.subtype :=
(indResAdjunction k S.subtype).ofNatIsoLeft (indCoindNatIso k S) | def | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | coindResAdjunction | Given a finite index subgroup `S ≤ G`, `Coind_S^G` is left adjoint to the restriction functor
`Res k G ⥤ Res k S`, since it is naturally isomorphic to `Ind_S^G`. |
@[simp]
coindResAdjunction_counit_app (B : Rep k G) :
(coindResAdjunction k S).counit.app B = (indCoindIso <| (Action.res _ S.subtype).obj B).inv ≫
(indResAdjunction k S.subtype).counit.app B := by
simp [coindResAdjunction, Adjunction.ofNatIsoLeft, Adjunction.equivHomsetLeftOfNatIso,
indResAdjunction]
@... | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | coindResAdjunction_counit_app | null |
coindResAdjunction_unit_app :
(coindResAdjunction k S).unit.app A = (indResAdjunction k S.subtype).unit.app A ≫
(Action.res _ S.subtype).map (indCoindIso A).hom := by
ext
simp [coindResAdjunction, Adjunction.ofNatIsoLeft, Adjunction.equivHomsetLeftOfNatIso,
indResAdjunction] | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | coindResAdjunction_unit_app | null |
coindResAdjunction_homEquiv_apply {B : Rep k G} (f : coind S.subtype A ⟶ B) :
(coindResAdjunction k S).homEquiv _ _ f =
indResHomEquiv S.subtype A B ((indCoindIso A).hom ≫ f) := by
simp only [coindResAdjunction, Adjunction.ofNatIsoLeft, indResAdjunction,
Adjunction.mkOfHomEquiv_homEquiv]
rfl | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | coindResAdjunction_homEquiv_apply | null |
coindResAdjunction_homEquiv_symm_apply
{B : Rep k G} (f : A ⟶ (Action.res _ S.subtype).obj B) :
((coindResAdjunction k S).homEquiv _ _).symm f =
(indCoindIso A).inv ≫ (indResHomEquiv S.subtype A B).symm f := by
simp only [coindResAdjunction, Adjunction.ofNatIsoLeft, indResAdjunction,
Adjunction.mkOf... | lemma | RepresentationTheory | [
"Mathlib.GroupTheory.Index",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/FiniteIndex.lean | coindResAdjunction_homEquiv_symm_apply | null |
IndV := Coinvariants (V := TensorProduct k (H →₀ k) A)
(Representation.tprod ((leftRegular k H).comp φ) ρ) | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | IndV | Given a group homomorphism `φ : G →* H` and a `G`-representation `(A, ρ)`, this is the
`k`-module `(k[H] ⊗[k] A)_G` with the `G`-representation on `k[H]` defined by `φ`.
See `Representation.ind` for the induced `H`-representation on `IndV φ ρ`. |
noncomputable IndV.mk (h : H) : A →ₗ[k] IndV φ ρ :=
Coinvariants.mk _ ∘ₗ TensorProduct.mk k _ _ (single h 1)
@[ext] | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | IndV.mk | Given a group homomorphism `φ : G →* H` and a `G`-representation `(A, ρ)`, this is the
`H → A →ₗ[k] (k[H] ⊗[k] A)_G` sending `h, a` to `⟦h ⊗ₜ a⟧`. |
IndV.hom_ext {f g : IndV φ ρ →ₗ[k] B}
(hfg : ∀ h : H, f ∘ₗ IndV.mk φ ρ h = g ∘ₗ IndV.mk φ ρ h) : f = g :=
Coinvariants.hom_ext <| TensorProduct.ext <| Finsupp.lhom_ext' fun h =>
LinearMap.ext_ring <| hfg h | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | IndV.hom_ext | null |
@[simps]
noncomputable ind : Representation k H (IndV φ ρ) where
toFun h := Coinvariants.map _ _ ((lmapDomain k k fun x => x * h⁻¹).rTensor _)
fun _ => by ext; simp [mul_assoc]
map_one' := by ext; simp
map_mul' _ _ := by ext; simp [IndV, mul_assoc] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | ind | Given a group homomorphism `φ : G →* H` and a `G`-representation `A`, this is
`(k[H] ⊗[k] A)_G` equipped with the `H`-representation defined by sending `h : H` and `⟦h₁ ⊗ₜ a⟧`
to `⟦h₁h⁻¹ ⊗ₜ a⟧`. |
ind_mk (h₁ h₂ : H) (a : A) :
ind φ ρ h₁ (IndV.mk _ _ h₂ a) = IndV.mk _ _ (h₂ * h₁⁻¹) a := by
simp | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | ind_mk | null |
noncomputable ind : Rep k H := Rep.of (A.ρ.ind φ) | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | ind | Given a group homomorphism `φ : G →* H` and a `G`-representation `A`, this is
`(k[H] ⊗[k] A)_G` equipped with the `H`-representation defined by sending `h : H` and `⟦h₁ ⊗ₜ a⟧`
to `⟦h₁h⁻¹ ⊗ₜ a⟧`. |
@[simps]
noncomputable indMap {A B : Rep k G} (f : A ⟶ B) : ind φ A ⟶ ind φ B where
hom := ModuleCat.ofHom <| Representation.Coinvariants.map _ _
(LinearMap.lTensor (H →₀ k) f.hom.hom) fun g => by ext; simp [hom_comm_apply]
comm _ := by
ext
simp [ModuleCat.endRingEquiv]
variable (k) in | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | indMap | Given a group homomorphism `φ : G →* H`, a morphism of `G`-representations `f : A ⟶ B` induces
a morphism of `H`-representations `(k[H] ⊗[k] A)_G ⟶ (k[H] ⊗[k] B)_G`. |
@[simps obj map]
noncomputable indFunctor : Rep k G ⥤ Rep k H where
obj A := ind φ A
map f := indMap φ f
map_id _ := by ext; rfl
map_comp _ _ := by ext; rfl | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | indFunctor | Given a group homomorphism `φ : G →* H`, this is the functor sending a `G`-representation `A`
to the induced `H`-representation `ind φ A`, with action on maps induced by left tensoring. |
@[simps]
noncomputable indResHomEquiv :
(ind φ A ⟶ B) ≃ₗ[k] (A ⟶ (Action.res _ φ).obj B) where
toFun f := {
hom := ModuleCat.ofHom (f.hom.hom ∘ₗ IndV.mk φ A.ρ 1)
comm g := by
ext x
have := (hom_comm_apply f (φ g) (IndV.mk φ A.ρ 1 x)).symm
simp_all [← Coinvariants.mk_inv_tmul] }
map_add... | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | indResHomEquiv | Given a group homomorphism `φ : G →* H`, an `H`-representation `B`, and a `G`-representation
`A`, there is a `k`-linear equivalence between the `H`-representation morphisms `ind φ A ⟶ B` and
the `G`-representation morphisms `A ⟶ B`. |
@[simps! unit_app_hom_hom counit_app_hom_hom]
noncomputable indResAdjunction : indFunctor k φ ⊣ Action.res _ φ :=
Adjunction.mkOfHomEquiv {
homEquiv A B := (indResHomEquiv φ A B).toEquiv
homEquiv_naturality_left_symm _ _ :=
Action.hom_ext _ _ <| ModuleCat.hom_ext <| IndV.hom_ext _ _ fun _ => by ext; sim... | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | indResAdjunction | Given a group homomorphism `φ : G →* H`, the induction functor `Rep k G ⥤ Rep k H` is left
adjoint to the restriction functor along `φ`. |
noncomputable coinvariantsTensorIndHom :
((coinvariantsTensor k H).obj (ind φ A)).obj B ⟶
((coinvariantsTensor k G).obj A).obj ((Action.res _ φ).obj B) :=
ModuleCat.ofHom <| Coinvariants.lift _ (TensorProduct.lift <|
Coinvariants.lift _ (TensorProduct.lift <| Finsupp.lift _ _ _
fun g => ((coinvari... | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | coinvariantsTensorIndHom | Given a group hom `φ : G →* H`, `A : Rep k G` and `B : Rep k H`, this is the `k`-linear map
`(Ind(φ)(A) ⊗ B))_H ⟶ (A ⊗ Res(φ)(B))_G` sending `⟦h ⊗ₜ a⟧ ⊗ₜ b` to `⟦a ⊗ ρ(h)(b)⟧` for all
`h : H`, `a : A`, and `b : B`. |
coinvariantsTensorIndHom_mk_tmul_indVMk (h : H) (x : A) (y : B) :
coinvariantsTensorIndHom φ A B (coinvariantsTensorMk _ _ (IndV.mk φ _ h x) y) =
coinvariantsTensorMk _ _ x (B.ρ h y) := by
simp [tensorObj_def, ModuleCat.MonoidalCategory.tensorObj, coinvariantsTensorIndHom,
coinvariantsTensorMk] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | coinvariantsTensorIndHom_mk_tmul_indVMk | null |
noncomputable coinvariantsTensorIndInv :
((coinvariantsTensor k G).obj A).obj ((Action.res _ φ).obj B) ⟶
((coinvariantsTensor k H).obj (ind φ A)).obj B :=
ModuleCat.ofHom <| Coinvariants.lift _ (TensorProduct.lift <|
(coinvariantsTensorMk (ind φ A) B) ∘ₗ IndV.mk _ _ 1)
fun s => by
simp only ... | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | coinvariantsTensorIndInv | Given a group hom `φ : G →* H`, `A : Rep k G` and `B : Rep k H`, this is the `k`-linear map
`(A ⊗ Res(φ)(B))_G ⟶ (Ind(φ)(A) ⊗ B))_H` sending `⟦a ⊗ₜ b⟧` to `⟦1 ⊗ₜ a⟧ ⊗ₜ b` for all
`a : A`, and `b : B`. |
coinvariantsTensorIndInv_mk_tmul_indMk (x : A) (y : B) :
coinvariantsTensorIndInv φ A B (Coinvariants.mk
(A.ρ.tprod (Rep.ρ ((Action.res _ φ).obj B))) <| x ⊗ₜ y) =
coinvariantsTensorMk _ _ (IndV.mk φ _ 1 x) y := by
simp [tensorObj_def, tensorObj, coinvariantsTensorIndInv, coinvariantsTensorMk] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | coinvariantsTensorIndInv_mk_tmul_indMk | null |
@[simps]
noncomputable coinvariantsTensorIndIso :
((coinvariantsTensor k H).obj (ind φ A)).obj B ≅
((coinvariantsTensor k G).obj A).obj ((Action.res _ φ).obj B) where
hom := coinvariantsTensorIndHom φ A B
inv := coinvariantsTensorIndInv φ A B
hom_inv_id := by
ext h a b
simpa [tensorObj_def, tens... | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | coinvariantsTensorIndIso | Given a group hom `φ : G →* H`, `A : Rep k G` and `B : Rep k H`, this is the `k`-linear
isomorphism `(Ind(φ)(A) ⊗ B))_H ⟶ (A ⊗ Res(φ)(B))_G` sending `⟦h ⊗ₜ a⟧ ⊗ₜ b` to `⟦a ⊗ ρ(h)(b)⟧`
for all `h : H`, `a : A`, and `b : B`. |
@[simps! hom_app inv_app]
noncomputable coinvariantsTensorIndNatIso :
(coinvariantsTensor k H).obj (ind φ A) ≅ Action.res _ φ ⋙ (coinvariantsTensor k G).obj A :=
NatIso.ofComponents (fun B => coinvariantsTensorIndIso φ A B) fun {X Y} f => by
ext
simp [tensorObj_def, tensorObj, coinvariantsTensorIndHom, co... | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Coinvariants"
] | Mathlib/RepresentationTheory/Induced.lean | coinvariantsTensorIndNatIso | Given a group hom `φ : G →* H` and `A : Rep k G`, the functor `Rep k H ⥤ ModuleCat k` sending
`B ↦ (Ind(φ)(A) ⊗ B))_H` is naturally isomorphic to the one sending `B ↦ (A ⊗ Res(φ)(B))_G`. |
noncomputable average : MonoidAlgebra k G :=
⅟(Fintype.card G : k) • ∑ g : G, of k G g | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | average | The average of all elements of the group `G`, considered as an element of `MonoidAlgebra k G`. |
@[simp]
mul_average_left (g : G) : ↑(Finsupp.single g 1) * average k G = average k G := by
simp only [mul_one, Finset.mul_sum, Algebra.mul_smul_comm, average, MonoidAlgebra.of_apply,
MonoidAlgebra.single_mul_single]
set f : G → MonoidAlgebra k G := fun x => Finsupp.single x 1
change ⅟(Fintype.card G : k) • ∑ ... | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | mul_average_left | `average k G` is invariant under left multiplication by elements of `G`. |
@[simp]
mul_average_right (g : G) : average k G * ↑(Finsupp.single g 1) = average k G := by
simp only [mul_one, Finset.sum_mul, Algebra.smul_mul_assoc, average, MonoidAlgebra.of_apply,
MonoidAlgebra.single_mul_single]
set f : G → MonoidAlgebra k G := fun x => Finsupp.single x 1
change ⅟(Fintype.card G : k) • ... | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | mul_average_right | `average k G` is invariant under right multiplication by elements of `G`. |
invariants : Submodule k V where
carrier := setOf fun v => ∀ g : G, ρ g v = v
zero_mem' g := by simp only [map_zero]
add_mem' hv hw g := by simp only [hv g, hw g, map_add]
smul_mem' r v hv g := by simp only [hv g, LinearMap.map_smulₛₗ, RingHom.id_apply]
@[simp] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | invariants | The subspace of invariants, consisting of the vectors fixed by all elements of `G`. |
mem_invariants (v : V) : v ∈ invariants ρ ↔ ∀ g : G, ρ g v = v := by rfl | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | mem_invariants | null |
invariants_eq_inter : (invariants ρ).carrier = ⋂ g : G, Function.fixedPoints (ρ g) := by
ext; simp [Function.IsFixedPt] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | invariants_eq_inter | null |
invariants_eq_top [ρ.IsTrivial] :
invariants ρ = ⊤ :=
eq_top_iff.2 (fun x _ g => ρ.isTrivial_apply g x) | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | invariants_eq_top | null |
mem_invariants_iff_of_forall_mem_zpowers
(g : G) (hg : ∀ x, x ∈ Subgroup.zpowers g) (x : V) :
x ∈ ρ.invariants ↔ ρ g x = x :=
⟨fun h => h g, fun hx γ => by
rcases hg γ with ⟨i, rfl⟩
induction i with | zero => simp | succ i _ => simp_all [zpow_add_one] | pred i h => _
simpa [neg_sub_comm _ (1 : ℤ),... | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | mem_invariants_iff_of_forall_mem_zpowers | null |
@[simp]
noncomputable averageMap : V →ₗ[k] V :=
asAlgebraHom ρ (average k G) | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | averageMap | The action of `average k G` gives a projection map onto the subspace of invariants. |
averageMap_invariant (v : V) : averageMap ρ v ∈ invariants ρ := fun g => by
rw [averageMap, ← asAlgebraHom_single_one, ← Module.End.mul_apply, ← map_mul (asAlgebraHom ρ),
mul_average_left] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | averageMap_invariant | The `averageMap` sends elements of `V` to the subspace of invariants. |
averageMap_id (v : V) (hv : v ∈ invariants ρ) : averageMap ρ v = v := by
rw [mem_invariants] at hv
simp [average, map_sum, hv, Finset.card_univ, ← Nat.cast_smul_eq_nsmul k _ v, smul_smul] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | averageMap_id | The `averageMap` acts as the identity on the subspace of invariants. |
isProj_averageMap : LinearMap.IsProj ρ.invariants ρ.averageMap :=
⟨ρ.averageMap_invariant, ρ.averageMap_id⟩ | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | isProj_averageMap | null |
le_comap_invariants (g : G) :
(invariants <| ρ.comp S.subtype) ≤
(invariants <| ρ.comp S.subtype).comap (ρ g) :=
fun x hx ⟨s, hs⟩ => by
simpa using congr(ρ g $(hx ⟨(g⁻¹ * s * g), Subgroup.Normal.conj_mem' ‹_› s hs g⟩)) | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | le_comap_invariants | null |
toInvariants :
Representation k G (invariants (ρ.comp S.subtype)) :=
subrepresentation ρ _ <| le_comap_invariants ρ S | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | toInvariants | Given a normal subgroup `S ≤ G`, a `G`-representation `ρ` restricts to a `G`-representation on
the invariants of `ρ|_S`. |
quotientToInvariants :
Representation k (G ⧸ S) (invariants (ρ.comp S.subtype)) :=
ofQuotient (toInvariants ρ S) S | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | quotientToInvariants | Given a normal subgroup `S ≤ G`, a `G`-representation `ρ` induces a `G ⧸ S`-representation on
the invariants of `ρ|_S`. |
mem_invariants_iff_comm {X Y : Rep k G} (f : X.V →ₗ[k] Y.V) (g : G) :
(linHom X.ρ Y.ρ) g f = f ↔ f.comp (X.ρ g) = (Y.ρ g).comp f := by
dsimp
rw [← LinearMap.comp_assoc, ← ModuleCat.hom_ofHom (Y.ρ g), ← ModuleCat.hom_ofHom f,
← ModuleCat.hom_comp, ← ModuleCat.hom_ofHom (X.ρ g⁻¹), ← ModuleCat.hom_comp,
... | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | mem_invariants_iff_comm | null |
@[simps]
invariantsEquivRepHom (X Y : Rep k G) : (linHom X.ρ Y.ρ).invariants ≃ₗ[k] X ⟶ Y where
toFun f := ⟨ModuleCat.ofHom f.val, fun g =>
ModuleCat.hom_ext ((mem_invariants_iff_comm _ g).1 (f.property g))⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨f.hom.hom, fun g =>
(mem_invariants_iff_com... | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | invariantsEquivRepHom | The invariants of the representation `linHom X.ρ Y.ρ` correspond to the representation
homomorphisms from `X` to `Y`. |
invariantsEquivFDRepHom (X Y : FDRep k G) : (linHom X.ρ Y.ρ).invariants ≃ₗ[k] X ⟶ Y := by
rw [← FDRep.forget₂_ρ, ← FDRep.forget₂_ρ]
exact linHom.invariantsEquivRepHom
((forget₂ (FDRep k G) (Rep k G)).obj X) ((forget₂ (FDRep k G) (Rep k G)).obj Y) ≪≫ₗ
FDRep.forget₂HomLinearEquiv X Y | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | invariantsEquivFDRepHom | The invariants of the representation `linHom X.ρ Y.ρ` correspond to the representation
homomorphisms from `X` to `Y`. |
toInvariants : Rep k G := Rep.of <| A.ρ.toInvariants S | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | toInvariants | Given a normal subgroup `S ≤ G`, a `G`-representation `ρ` restricts to a `G`-representation on
the invariants of `ρ|_S`. |
quotientToInvariants : Rep k (G ⧸ S) := Rep.of (A.ρ.quotientToInvariants S)
variable (k G) | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | quotientToInvariants | Given a normal subgroup `S ≤ G`, a `G`-representation `ρ` induces a `G ⧸ S`-representation on
the invariants of `ρ|_S`. |
@[simps! obj_carrier map_hom]
noncomputable invariantsFunctor : Rep k G ⥤ ModuleCat k where
obj A := ModuleCat.of k A.ρ.invariants
map {A B} f := ModuleCat.ofHom <| (f.hom.hom ∘ₗ A.ρ.invariants.subtype).codRestrict
B.ρ.invariants fun ⟨c, hc⟩ g => by
have := (hom_comm_apply f g c).symm
simp_all [hc g... | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | invariantsFunctor | The functor sending a representation to its submodule of invariants. |
@[simps obj_V map_hom]
noncomputable quotientToInvariantsFunctor (S : Subgroup G) [S.Normal] :
Rep k G ⥤ Rep k (G ⧸ S) where
obj X := X.quotientToInvariants S
map {X Y} f := {
hom := (invariantsFunctor k S).map ((Action.res _ S.subtype).map f)
comm g := QuotientGroup.induction_on g fun g => by
ext... | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | quotientToInvariantsFunctor | Given a normal subgroup S ≤ G, this is the functor sending a `G`-representation `A` to the
`G ⧸ S`-representation it induces on `A^S`. |
@[simps]
noncomputable invariantsAdjunction : trivialFunctor k G ⊣ invariantsFunctor k G where
unit := { app _ := ModuleCat.ofHom <| LinearMap.id.codRestrict _ <| by simp [trivialFunctor] }
counit := { app X := {
hom := ModuleCat.ofHom <| Submodule.subtype _
comm g := by ext x; exact (x.2 g).symm }}
@[simp] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | invariantsAdjunction | The adjunction between the functor equipping a module with the trivial representation, and
the functor sending a representation to its submodule of invariants. |
invariantsAdjunction_homEquiv_apply_hom
{X : ModuleCat k} {Y : Rep k G} (f : (trivialFunctor k G).obj X ⟶ Y) :
((invariantsAdjunction k G).homEquiv _ _ f).hom =
f.hom.hom.codRestrict _ (by intro _ _; exact (hom_comm_apply f _ _).symm) := rfl
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | invariantsAdjunction_homEquiv_apply_hom | null |
invariantsAdjunction_homEquiv_symm_apply_hom
{X : ModuleCat k} {Y : Rep k G} (f : X ⟶ (invariantsFunctor k G).obj Y) :
(((invariantsAdjunction k G).homEquiv _ _).symm f).hom.hom =
Submodule.subtype _ ∘ₗ f.hom := rfl | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Basic",
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Invariants.lean | invariantsAdjunction_homEquiv_symm_apply_hom | null |
conjugate (g : G) : W →ₗ[k] V :=
GroupSMul.linearMap k V g⁻¹ ∘ₗ π ∘ₗ GroupSMul.linearMap k W g | def | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | conjugate | We define the conjugate of `π` by `g`, as a `k`-linear map. |
conjugate_apply (g : G) (v : W) :
π.conjugate g v = MonoidAlgebra.single g⁻¹ (1 : k) • π (MonoidAlgebra.single g (1 : k) • v) :=
rfl
variable (i : V →ₗ[MonoidAlgebra k G] W) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | conjugate_apply | null |
conjugate_i (h : ∀ v : V, π (i v) = v) (g : G) (v : V) :
(conjugate π g : W → V) (i v) = v := by
rw [conjugate_apply, ← i.map_smul, h, ← mul_smul, single_mul_single, mul_one, inv_mul_cancel,
← one_def, one_smul] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | conjugate_i | null |
sumOfConjugates : W →ₗ[k] V :=
∑ g : G, π.conjugate g | def | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | sumOfConjugates | The sum of the conjugates of `π` by each element `g : G`, as a `k`-linear map.
(We postpone dividing by the size of the group as long as possible.) |
sumOfConjugates_apply (v : W) : π.sumOfConjugates G v = ∑ g : G, π.conjugate g v :=
LinearMap.sum_apply _ _ _ | lemma | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | sumOfConjugates_apply | null |
sumOfConjugatesEquivariant : W →ₗ[MonoidAlgebra k G] V :=
MonoidAlgebra.equivariantOfLinearOfComm (π.sumOfConjugates G) fun g v => by
simp only [sumOfConjugates_apply, Finset.smul_sum, conjugate_apply]
refine Fintype.sum_bijective (· * g) (Group.mulRight_bijective g) _ _ fun i ↦ ?_
simp only [smul_smul, s... | def | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | sumOfConjugatesEquivariant | In fact, the sum over `g : G` of the conjugate of `π` by `g` is a `k[G]`-linear map. |
sumOfConjugatesEquivariant_apply (v : W) :
π.sumOfConjugatesEquivariant G v = ∑ g : G, π.conjugate g v :=
π.sumOfConjugates_apply G v | theorem | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | sumOfConjugatesEquivariant_apply | null |
equivariantProjection : W →ₗ[MonoidAlgebra k G] V :=
Ring.inverse (Fintype.card G : k) • π.sumOfConjugatesEquivariant G | def | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | equivariantProjection | We construct our `k[G]`-linear retraction of `i` as
$$ \frac{1}{|G|} \sum_{g \in G} g⁻¹ • π(g • -). $$ |
equivariantProjection_apply (v : W) :
π.equivariantProjection G v = Ring.inverse (Fintype.card G : k) • ∑ g : G, π.conjugate g v := by
simp only [equivariantProjection, smul_apply, sumOfConjugatesEquivariant_apply] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | equivariantProjection_apply | null |
equivariantProjection_condition (hcard : IsUnit (Fintype.card G : k))
(h : ∀ v : V, π (i v) = v) (v : V) : (π.equivariantProjection G) (i v) = v := by
rw [equivariantProjection_apply]
simp only [conjugate_i π i h]
rw [Finset.sum_const, Finset.card_univ, ← Nat.cast_smul_eq_nsmul k, smul_smul,
Ring.inverse_... | theorem | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | equivariantProjection_condition | null |
exists_leftInverse_of_injective
(f : V →ₗ[MonoidAlgebra k G] W) (hf : LinearMap.ker f = ⊥) :
∃ g : W →ₗ[MonoidAlgebra k G] V, g.comp f = LinearMap.id := by
let A := MonoidAlgebra k G
letI : Module k W := .compHom W (algebraMap k A)
letI : Module k V := .compHom V (algebraMap k A)
have := IsScalarTower.o... | theorem | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | exists_leftInverse_of_injective | null |
exists_isCompl (p : Submodule (MonoidAlgebra k G) V) :
∃ q : Submodule (MonoidAlgebra k G) V, IsCompl p q := by
rcases MonoidAlgebra.exists_leftInverse_of_injective p.subtype p.ker_subtype with ⟨f, hf⟩
exact ⟨LinearMap.ker f, LinearMap.isCompl_of_proj <| DFunLike.congr_fun hf⟩ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Group.TypeTags.Finite",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/RepresentationTheory/Maschke.lean | exists_isCompl | null |
Rep (k G : Type u) [Ring k] [Monoid G] :=
Action (ModuleCat.{u} k) G | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | Rep | The category of `k`-linear representations of a monoid `G`. |
ρ (V : Rep k G) : Representation k G V :=
(ModuleCat.endRingEquiv V.V).toMonoidHom.comp (Action.ρ V) | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | ρ | Specialize the existing `Action.ρ`, changing the type to `Representation k G V`. |
of {V : Type u} [AddCommGroup V] [Module k V] (ρ : G →* V →ₗ[k] V) : Rep k G :=
⟨ModuleCat.of k V, (ModuleCat.endRingEquiv _).symm.toMonoidHom.comp ρ⟩ | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | of | Lift an unbundled representation to `Rep`. |
coe_of {V : Type u} [AddCommGroup V] [Module k V] (ρ : G →* V →ₗ[k] V) :
(of ρ : Type u) = V :=
rfl
@[simp] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | coe_of | null |
of_ρ {V : Type u} [AddCommGroup V] [Module k V] (ρ : G →* V →ₗ[k] V) : (of ρ).ρ = ρ :=
rfl | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | of_ρ | null |
Action_ρ_eq_ρ {A : Rep k G} :
Action.ρ A = (ModuleCat.endRingEquiv _).symm.toMonoidHom.comp A.ρ :=
rfl
@[simp] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | Action_ρ_eq_ρ | null |
ρ_hom {X : Rep k G} (g : G) : (Action.ρ X g).hom = X.ρ g := rfl
@[simp] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | ρ_hom | null |
ofHom_ρ {X : Rep k G} (g : G) : ModuleCat.ofHom (X.ρ g) = Action.ρ X g := rfl
@[deprecated Representation.inv_self_apply (since := "2025-05-09")] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | ofHom_ρ | null |
ρ_inv_self_apply {G : Type u} [Group G] (A : Rep k G) (g : G) (x : A) :
A.ρ g⁻¹ (A.ρ g x) = x :=
show (A.ρ g⁻¹ * A.ρ g) x = x by rw [← map_mul, inv_mul_cancel, map_one, Module.End.one_apply]
@[deprecated Representation.self_inv_apply (since := "2025-05-09")] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | ρ_inv_self_apply | null |
ρ_self_inv_apply {G : Type u} [Group G] {A : Rep k G} (g : G) (x : A) :
A.ρ g (A.ρ g⁻¹ x) = x :=
show (A.ρ g * A.ρ g⁻¹) x = x by rw [← map_mul, mul_inv_cancel, map_one, Module.End.one_apply] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | ρ_self_inv_apply | null |
hom_comm_apply {A B : Rep k G} (f : A ⟶ B) (g : G) (x : A) :
f.hom (A.ρ g x) = B.ρ g (f.hom x) :=
LinearMap.ext_iff.1 (ModuleCat.hom_ext_iff.mp (f.comm g)) x
variable (k G) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | hom_comm_apply | null |
trivial (V : Type u) [AddCommGroup V] [Module k V] : Rep k G :=
Rep.of (Representation.trivial k G V)
variable {k G} | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | trivial | The trivial `k`-linear `G`-representation on a `k`-module `V.` |
trivial_def {V : Type u} [AddCommGroup V] [Module k V] (g : G) :
(trivial k G V).ρ g = LinearMap.id :=
rfl
variable (k G) in | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | trivial_def | null |
@[simps! obj_V map_hom]
trivialFunctor : ModuleCat k ⥤ Rep k G where
obj V := trivial k G V
map f := { hom := f, comm := fun _ => rfl } | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | trivialFunctor | The functor equipping a module with the trivial representation. |
IsTrivial (A : Rep k G) := A.ρ.IsTrivial | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | IsTrivial | A predicate for representations that fix every element. |
@[simps]
applyAsHom (g : G) : A ⟶ A where
hom := ModuleCat.ofHom (A.ρ g)
comm _ := by ext; simp [← Module.End.mul_apply, ← map_mul, mul_comm]
@[reassoc, elementwise] | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | applyAsHom | Given a representation `A` of a commutative monoid `G`, the map `ρ_A(g)` is a representation
morphism `A ⟶ A` for any `g : G`. |
applyAsHom_comm {A B : Rep k G} (f : A ⟶ B) (g : G) :
A.applyAsHom g ≫ f = f ≫ B.applyAsHom g := by
ext
simp [hom_comm_apply] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | applyAsHom_comm | null |
ofQuotient : Rep k (G ⧸ S) := Rep.of (A.ρ.ofQuotient S) | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | ofQuotient | Given a normal subgroup `S ≤ G`, a `G`-representation `ρ` which is trivial on `S` factors
through `G ⧸ S`. |
resOfQuotientIso [Representation.IsTrivial (A.ρ.comp S.subtype)] :
(Action.res _ (QuotientGroup.mk' S)).obj (A.ofQuotient S) ≅ A := Iso.refl _ | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | resOfQuotientIso | A `G`-representation `A` on which a normal subgroup `S ≤ G` acts trivially induces a
`G ⧸ S`-representation on `A`, and composing this with the quotient map `G → G ⧸ S` gives the
original representation by definition. Useful for typechecking. |
subrepresentation (W : Submodule k A) (le_comap : ∀ g, W ≤ W.comap (A.ρ g)) :
Rep k G :=
Rep.of (A.ρ.subrepresentation W le_comap) | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | subrepresentation | Given a `k`-linear `G`-representation `(V, ρ)`, this is the representation defined by
restricting `ρ` to a `G`-invariant `k`-submodule of `V`. |
@[simps]
subtype (W : Submodule k A) (le_comap : ∀ g, W ≤ W.comap (A.ρ g)) :
subrepresentation A W le_comap ⟶ A where
hom := ModuleCat.ofHom W.subtype
comm _ := rfl | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | subtype | The natural inclusion of a subrepresentation into the ambient representation. |
quotient (W : Submodule k A) (le_comap : ∀ g, W ≤ W.comap (A.ρ g)) :
Rep k G :=
Rep.of (A.ρ.quotient W le_comap) | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | quotient | Given a `k`-linear `G`-representation `(V, ρ)` and a `G`-invariant `k`-submodule `W ≤ V`, this
is the representation induced on `V ⧸ W` by `ρ`. |
@[simps]
mkQ (W : Submodule k A) (le_comap : ∀ g, W ≤ W.comap (A.ρ g)) :
A ⟶ quotient A W le_comap where
hom := ModuleCat.ofHom <| Submodule.mkQ _
comm _ := rfl | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | mkQ | The natural projection from a representation to its quotient by a subrepresentation. |
epi_iff_surjective {A B : Rep k G} (f : A ⟶ B) : Epi f ↔ Function.Surjective f.hom :=
⟨fun _ => (ModuleCat.epi_iff_surjective ((forget₂ _ _).map f)).1 inferInstance,
fun h => (forget₂ _ _).epi_of_epi_map ((ModuleCat.epi_iff_surjective <|
(forget₂ _ _).map f).2 h)⟩ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | epi_iff_surjective | null |
mono_iff_injective {A B : Rep k G} (f : A ⟶ B) : Mono f ↔ Function.Injective f.hom :=
⟨fun _ => (ModuleCat.mono_iff_injective ((forget₂ _ _).map f)).1 inferInstance,
fun h => (forget₂ _ _).mono_of_mono_map ((ModuleCat.mono_iff_injective <|
(forget₂ _ _).map f).2 h)⟩ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | mono_iff_injective | null |
@[simp]
tensor_ρ {A B : Rep k G} : (A ⊗ B).ρ = A.ρ.tprod B.ρ := rfl
@[simp] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Adjunctions",
"Mathlib.Algebra.Category.ModuleCat.EpiMono",
"Mathlib.Algebra.Category.ModuleCat.Limits",
"Mathlib.Algebra.Category.ModuleCat.Colimits",
"Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric",
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.C... | Mathlib/RepresentationTheory/Rep.lean | tensor_ρ | null |
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