statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
forget_braided_faithful : faithful (forget_braided V G).to_functor | by { change faithful (forget V G), apply_instance, } | instance | Action.forget_braided_faithful | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_monoidal_equivalence : monoidal_functor (Action V G) (single_obj G ⥤ V) | monoidal.from_transported (Action.functor_category_equivalence _ _).symm | def | Action.functor_category_monoidal_equivalence | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Action.functor_category_equivalence"
] | Upgrading the functor `Action V G ⥤ (single_obj G ⥤ V)` to a monoidal functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_category_monoidal_equivalence.μ_app (A B : Action V G) :
((functor_category_monoidal_equivalence V G).μ A B).app punit.star = 𝟙 _ | begin
dunfold functor_category_monoidal_equivalence,
simp only [monoidal.from_transported_to_lax_monoidal_functor_μ],
show (𝟙 A.V ⊗ 𝟙 B.V) ≫ 𝟙 (A.V ⊗ B.V) ≫ (𝟙 A.V ⊗ 𝟙 B.V) = 𝟙 (A.V ⊗ B.V),
simp only [monoidal_category.tensor_id, category.comp_id],
end | lemma | Action.functor_category_monoidal_equivalence.μ_app | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_monoidal_equivalence.μ_iso_inv_app (A B : Action V G) :
((functor_category_monoidal_equivalence V G).μ_iso A B).inv.app punit.star = 𝟙 _ | begin
rw [←nat_iso.app_inv, ←is_iso.iso.inv_hom],
refine is_iso.inv_eq_of_hom_inv_id _,
rw [category.comp_id, nat_iso.app_hom, monoidal_functor.μ_iso_hom,
functor_category_monoidal_equivalence.μ_app],
end | lemma | Action.functor_category_monoidal_equivalence.μ_iso_inv_app | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_monoidal_equivalence.ε_app :
(functor_category_monoidal_equivalence V G).ε.app punit.star = 𝟙 _ | begin
dunfold functor_category_monoidal_equivalence,
simp only [monoidal.from_transported_to_lax_monoidal_functor_ε],
show 𝟙 (monoidal_category.tensor_unit V) ≫ _ = 𝟙 (monoidal_category.tensor_unit V),
rw [nat_iso.is_iso_inv_app, category.id_comp],
exact is_iso.inv_id,
end | lemma | Action.functor_category_monoidal_equivalence.ε_app | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_monoidal_equivalence.inv_counit_app_hom (A : Action V G) :
((functor_category_monoidal_equivalence _ _).inv.adjunction.counit.app A).hom = 𝟙 _ | rfl | lemma | Action.functor_category_monoidal_equivalence.inv_counit_app_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_monoidal_equivalence.counit_app (A : single_obj G ⥤ V) :
((functor_category_monoidal_equivalence _ _).adjunction.counit.app A).app punit.star = 𝟙 _ | rfl | lemma | Action.functor_category_monoidal_equivalence.counit_app | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_monoidal_equivalence.inv_unit_app_app
(A : single_obj G ⥤ V) :
((functor_category_monoidal_equivalence _ _).inv.adjunction.unit.app A).app
punit.star = 𝟙 _ | rfl | lemma | Action.functor_category_monoidal_equivalence.inv_unit_app_app | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_monoidal_equivalence.unit_app_hom (A : Action V G) :
((functor_category_monoidal_equivalence _ _).adjunction.unit.app A).hom = 𝟙 _ | rfl | lemma | Action.functor_category_monoidal_equivalence.unit_app_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_monoidal_equivalence.functor_map {A B : Action V G} (f : A ⟶ B) :
(functor_category_monoidal_equivalence _ _).map f
= functor_category_equivalence.functor.map f | rfl | lemma | Action.functor_category_monoidal_equivalence.functor_map | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_monoidal_equivalence.inverse_map
{A B : single_obj G ⥤ V} (f : A ⟶ B) :
(functor_category_monoidal_equivalence _ _).inv.map f
= functor_category_equivalence.inverse.map f | rfl | lemma | Action.functor_category_monoidal_equivalence.inverse_map | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_dual_V [right_rigid_category V] : (Xᘁ).V = (X.V)ᘁ | rfl | lemma | Action.right_dual_V | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_dual_V [left_rigid_category V] : (ᘁX).V = ᘁ(X.V) | rfl | lemma | Action.left_dual_V | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_dual_ρ [right_rigid_category V] (h : H) : (Xᘁ).ρ h = (X.ρ (h⁻¹ : H))ᘁ | by { rw ←single_obj.inv_as_inv, refl } | lemma | Action.right_dual_ρ | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_dual_ρ [left_rigid_category V] (h : H) : (ᘁX).ρ h = ᘁ(X.ρ (h⁻¹ : H)) | by { rw ←single_obj.inv_as_inv, refl } | lemma | Action.left_dual_ρ | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Action_punit_equivalence : Action V (Mon.of punit) ≌ V | { functor := forget V _,
inverse :=
{ obj := λ X, ⟨X, 1⟩,
map := λ X Y f, ⟨f, λ ⟨⟩, by simp⟩, },
unit_iso := nat_iso.of_components (λ X, mk_iso (iso.refl _) (λ ⟨⟩, by simpa using ρ_one X))
(by tidy),
counit_iso := nat_iso.of_components (λ X, iso.refl _) (by tidy), } | def | Action.Action_punit_equivalence | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Mon.of"
] | Actions/representations of the trivial group are just objects in the ambient category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
res {G H : Mon} (f : G ⟶ H) : Action V H ⥤ Action V G | { obj := λ M,
{ V := M.V,
ρ := f ≫ M.ρ },
map := λ M N p,
{ hom := p.hom,
comm' := λ g, p.comm (f g) } } | def | Action.res | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Mon"
] | The "restriction" functor along a monoid homomorphism `f : G ⟶ H`,
taking actions of `H` to actions of `G`.
(This makes sense for any homomorphism, but the name is natural when `f` is a monomorphism.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
res_id {G : Mon} : res V (𝟙 G) ≅ 𝟭 (Action V G) | nat_iso.of_components (λ M, mk_iso (iso.refl _) (by tidy)) (by tidy) | def | Action.res_id | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Mon"
] | The natural isomorphism from restriction along the identity homomorphism to
the identity functor on `Action V G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
res_comp {G H K : Mon} (f : G ⟶ H) (g : H ⟶ K) : res V g ⋙ res V f ≅ res V (f ≫ g) | nat_iso.of_components (λ M, mk_iso (iso.refl _) (by tidy)) (by tidy) | def | Action.res_comp | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Mon"
] | The natural isomorphism from the composition of restrictions along homomorphisms
to the restriction along the composition of homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
res_additive [preadditive V] : (res V f).additive | {} | instance | Action.res_additive | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
res_linear [preadditive V] [linear R V] : (res V f).linear R | {} | instance | Action.res_linear | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_action (G H : Type u) [monoid G] [mul_action G H] : Action (Type u) (Mon.of G) | { V := H,
ρ := @mul_action.to_End_hom _ _ _ (by assumption) } | def | Action.of_mul_action | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Mon.of",
"monoid",
"mul_action",
"mul_action.to_End_hom"
] | Bundles a type `H` with a multiplicative action of `G` as an `Action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_mul_action_apply {G H : Type u} [monoid G] [mul_action G H] (g : G) (x : H) :
(of_mul_action G H).ρ g x = (g • x : H) | rfl | lemma | Action.of_mul_action_apply | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"monoid",
"mul_action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_action_limit_cone {ι : Type v} (G : Type (max v u)) [monoid G]
(F : ι → Type (max v u)) [Π i : ι, mul_action G (F i)] :
limit_cone (discrete.functor (λ i : ι, Action.of_mul_action G (F i))) | { cone :=
{ X := Action.of_mul_action G (Π i : ι, F i),
π :=
{ app := λ i, ⟨λ x, x i.as, λ g, by ext; refl⟩,
naturality' := λ i j x,
begin
ext,
discrete_cases,
cases x,
congr
end } },
is_limit :=
{ lift := λ s,
{ hom := λ x i, (s.π.app ⟨i⟩).hom x,
... | def | Action.of_mul_action_limit_cone | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action.of_mul_action",
"comm",
"lift",
"monoid",
"mul_action"
] | Given a family `F` of types with `G`-actions, this is the limit cone demonstrating that the
product of `F` as types is a product in the category of `G`-sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_regular (G : Type u) [monoid G] : Action (Type u) (Mon.of G) | Action.of_mul_action G G | def | Action.left_regular | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Action.of_mul_action",
"Mon.of",
"monoid"
] | The `G`-set `G`, acting on itself by left multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal (G : Type u) [monoid G] (n : ℕ) : Action (Type u) (Mon.of G) | Action.of_mul_action G (fin n → G) | def | Action.diagonal | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Action.of_mul_action",
"Mon.of",
"monoid"
] | The `G`-set `Gⁿ`, acting on itself by left multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal_one_iso_left_regular (G : Type u) [monoid G] :
diagonal G 1 ≅ left_regular G | Action.mk_iso (equiv.fun_unique _ _).to_iso (λ g, rfl) | def | Action.diagonal_one_iso_left_regular | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action.mk_iso",
"equiv.fun_unique",
"monoid"
] | We have `fin 1 → G ≅ G` as `G`-sets, with `G` acting by left multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_regular_tensor_iso (G : Type u) [group G]
(X : Action (Type u) (Mon.of G)) :
left_regular G ⊗ X ≅ left_regular G ⊗ Action.mk X.V 1 | { hom :=
{ hom := λ g, ⟨g.1, (X.ρ (g.1⁻¹ : G) g.2 : X.V)⟩,
comm' := λ g, funext $ λ x, prod.ext rfl $
show (X.ρ ((g * x.1)⁻¹ : G) * X.ρ g) x.2 = _,
by simpa only [mul_inv_rev, ←X.ρ.map_mul, inv_mul_cancel_right] },
inv :=
{ hom := λ g, ⟨g.1, X.ρ g.1 g.2⟩,
comm' := λ g, funext $ λ x, prod.ext r... | def | Action.left_regular_tensor_iso | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Mon.of",
"group",
"inv_mul_cancel_right",
"inv_mul_self",
"map_mul",
"mul_inv_rev",
"mul_inv_self",
"prod.ext"
] | Given `X : Action (Type u) (Mon.of G)` for `G` a group, then `G × X` (with `G` acting as left
multiplication on the first factor and by `X.ρ` on the second) is isomorphic as a `G`-set to
`G × X` (with `G` acting as left multiplication on the first factor and trivially on the second).
The isomorphism is given by `(g, x)... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal_succ (G : Type u) [monoid G] (n : ℕ) :
diagonal G (n + 1) ≅ left_regular G ⊗ diagonal G n | mk_iso (equiv.pi_fin_succ_above_equiv _ 0).to_iso (λ g, rfl) | def | Action.diagonal_succ | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"equiv.pi_fin_succ_above_equiv",
"monoid"
] | The natural isomorphism of `G`-sets `Gⁿ⁺¹ ≅ G × Gⁿ`, where `G` acts by left multiplication on
each factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_Action (F : V ⥤ W) (G : Mon.{u}) : Action V G ⥤ Action W G | { obj := λ M,
{ V := F.obj M.V,
ρ :=
{ to_fun := λ g, F.map (M.ρ g),
map_one' := by simp only [End.one_def, Action.ρ_one, F.map_id],
map_mul' := λ g h, by simp only [End.mul_def, F.map_comp, map_mul], }, },
map := λ M N f,
{ hom := F.map f.hom,
comm' := λ g, by { dsimp, rw [←F.map_comp, f.... | def | category_theory.functor.map_Action | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action",
"Action.comp_hom",
"Action.id_hom",
"Action.ρ_one",
"map_mul"
] | A functor between categories induces a functor between
the categories of `G`-actions within those categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_Action_preadditive [F.additive] : (F.map_Action G).additive | {} | instance | category_theory.functor.map_Action_preadditive | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_Action_linear [F.additive] [F.linear R] : (F.map_Action G).linear R | {} | instance | category_theory.functor.map_Action_linear | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_Action :
monoidal_functor (Action V G) (Action W G) | { ε :=
{ hom := F.ε,
comm' := λ g,
by { dsimp, erw [category.id_comp, category_theory.functor.map_id, category.comp_id], }, },
μ := λ X Y,
{ hom := F.μ X.V Y.V,
comm' := λ g, F.to_lax_monoidal_functor.μ_natural (X.ρ g) (Y.ρ g), },
ε_is_iso := by apply_instance,
μ_is_iso := by apply_instance,
μ_n... | def | category_theory.monoidal_functor.map_Action | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | A monoidal functor induces a monoidal functor between
the categories of `G`-actions within those categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_Action_ε_inv_hom :
(inv (F.map_Action G).ε).hom = inv F.ε | begin
ext,
simp only [←F.map_Action_to_lax_monoidal_functor_ε_hom G, ←Action.comp_hom,
is_iso.hom_inv_id, id_hom],
end | lemma | category_theory.monoidal_functor.map_Action_ε_inv_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_Action_μ_inv_hom (X Y : Action V G) :
(inv ((F.map_Action G).μ X Y)).hom = inv (F.μ X.V Y.V) | begin
ext,
simpa only [←F.map_Action_to_lax_monoidal_functor_μ_hom G, ←Action.comp_hom,
is_iso.hom_inv_id, id_hom],
end | lemma | category_theory.monoidal_functor.map_Action_μ_inv_hom | representation_theory | src/representation_theory/Action.lean | [
"algebra.category.Group.basic",
"category_theory.single_obj",
"category_theory.limits.functor_category",
"category_theory.limits.preserves.basic",
"category_theory.adjunction.limits",
"category_theory.monoidal.functor_category",
"category_theory.monoidal.transport",
"category_theory.monoidal.rigid.of_... | [
"Action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
representation | G →* (V →ₗ[k] V) | abbreviation | representation | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [] | A representation of `G` on the `k`-module `V` is an homomorphism `G →* (V →ₗ[k] V)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivial : representation k G V | 1 | def | representation.trivial | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"representation"
] | The trivial representation of `G` on a `k`-module V. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivial_def (g : G) (v : V) : trivial k g v = v | rfl | lemma | representation.trivial_def | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_algebra_hom : monoid_algebra k G →ₐ[k] (module.End k V) | (lift k G _) ρ | def | representation.as_algebra_hom | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"lift",
"module.End",
"monoid_algebra"
] | A `k`-linear representation of `G` on `V` can be thought of as
an algebra map from `monoid_algebra k G` into the `k`-linear endomorphisms of `V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_algebra_hom_def : as_algebra_hom ρ = (lift k G _) ρ | rfl | lemma | representation.as_algebra_hom_def | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_algebra_hom_single (g : G) (r : k) :
(as_algebra_hom ρ (finsupp.single g r)) = r • ρ g | by simp only [as_algebra_hom_def, monoid_algebra.lift_single] | lemma | representation.as_algebra_hom_single | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"finsupp.single",
"monoid_algebra.lift_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_algebra_hom_single_one (g : G):
(as_algebra_hom ρ (finsupp.single g 1)) = ρ g | by simp | lemma | representation.as_algebra_hom_single_one | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"finsupp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_algebra_hom_of (g : G) :
(as_algebra_hom ρ (of k G g)) = ρ g | by simp only [monoid_algebra.of_apply, as_algebra_hom_single, one_smul] | lemma | representation.as_algebra_hom_of | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_module (ρ : representation k G V) | V | def | representation.as_module | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"representation"
] | If `ρ : representation k G V`, then `ρ.as_module` is a type synonym for `V`,
which we equip with an instance `module (monoid_algebra k G) ρ.as_module`.
You should use `as_module_equiv : ρ.as_module ≃+ V` to translate terms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_module_module : module (monoid_algebra k G) ρ.as_module | module.comp_hom V (as_algebra_hom ρ).to_ring_hom | instance | representation.as_module_module | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"module",
"module.comp_hom",
"monoid_algebra"
] | A `k`-linear representation of `G` on `V` can be thought of as
a module over `monoid_algebra k G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_module_equiv : ρ.as_module ≃+ V | add_equiv.refl _ | def | representation.as_module_equiv | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [] | The additive equivalence from the `module (monoid_algebra k G)` to the original vector space
of the representative.
This is just the identity, but it is helpful for typechecking and keeping track of instances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_module_equiv_map_smul (r : monoid_algebra k G) (x : ρ.as_module) :
ρ.as_module_equiv (r • x) = ρ.as_algebra_hom r (ρ.as_module_equiv x) | rfl | lemma | representation.as_module_equiv_map_smul | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"monoid_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_module_equiv_symm_map_smul (r : k) (x : V) :
ρ.as_module_equiv.symm (r • x) =
algebra_map k (monoid_algebra k G) r • (ρ.as_module_equiv.symm x) | begin
apply_fun ρ.as_module_equiv,
simp,
end | lemma | representation.as_module_equiv_symm_map_smul | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"algebra_map",
"monoid_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_module_equiv_symm_map_rho (g : G) (x : V) :
ρ.as_module_equiv.symm (ρ g x) = monoid_algebra.of k G g • (ρ.as_module_equiv.symm x) | begin
apply_fun ρ.as_module_equiv,
simp,
end | lemma | representation.as_module_equiv_symm_map_rho | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"monoid_algebra.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_module' (M : Type*) [add_comm_monoid M] [module k M] [module (monoid_algebra k G) M]
[is_scalar_tower k (monoid_algebra k G) M] : representation k G M | (monoid_algebra.lift k G (M →ₗ[k] M)).symm (algebra.lsmul k M) | def | representation.of_module' | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"add_comm_monoid",
"algebra.lsmul",
"is_scalar_tower",
"module",
"monoid_algebra",
"monoid_algebra.lift",
"representation"
] | Build a `representation k G M` from a `[module (monoid_algebra k G) M]`.
This version is not always what we want, as it relies on an existing `[module k M]`
instance, along with a `[is_scalar_tower k (monoid_algebra k G) M]` instance.
We remedy this below in `of_module`
(with the tradeoff that the representation is d... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_module :
representation k G (restrict_scalars k (monoid_algebra k G) M) | (monoid_algebra.lift k G
(restrict_scalars k (monoid_algebra k G) M →ₗ[k] restrict_scalars k (monoid_algebra k G) M)).symm
(restrict_scalars.lsmul k (monoid_algebra k G) M) | def | representation.of_module | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"monoid_algebra",
"monoid_algebra.lift",
"representation",
"restrict_scalars",
"restrict_scalars.lsmul"
] | Build a `representation` from a `[module (monoid_algebra k G) M]`.
Note that the representation is built on `restrict_scalars k (monoid_algebra k G) M`,
rather than on `M` itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_module_as_algebra_hom_apply_apply
(r : monoid_algebra k G) (m : restrict_scalars k (monoid_algebra k G) M) :
((((of_module k G M).as_algebra_hom) r) m) =
(restrict_scalars.add_equiv _ _ _).symm (r • restrict_scalars.add_equiv _ _ _ m) | begin
apply monoid_algebra.induction_on r,
{ intros g,
simp only [one_smul, monoid_algebra.lift_symm_apply, monoid_algebra.of_apply,
representation.as_algebra_hom_single, representation.of_module,
add_equiv.apply_eq_iff_eq, restrict_scalars.lsmul_apply_apply], },
{ intros f g fw gw,
simp only ... | lemma | representation.of_module_as_algebra_hom_apply_apply | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"add_smul",
"alg_hom.map_smul",
"linear_map.add_apply",
"linear_map.smul_apply",
"monoid_algebra",
"monoid_algebra.induction_on",
"monoid_algebra.lift_symm_apply",
"one_smul",
"representation.as_algebra_hom_single",
"representation.of_module",
"restrict_scalars",
"restrict_scalars.add_equiv",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_module_as_module_act (g : G) (x : restrict_scalars k (monoid_algebra k G) ρ.as_module) :
of_module k G (ρ.as_module) g x =
(restrict_scalars.add_equiv _ _ _).symm ((ρ.as_module_equiv).symm
(ρ g (ρ.as_module_equiv (restrict_scalars.add_equiv _ _ _ x)))) | begin
apply_fun restrict_scalars.add_equiv _ _ ρ.as_module using
(restrict_scalars.add_equiv _ _ _).injective,
dsimp [of_module, restrict_scalars.lsmul_apply_apply],
simp,
end | lemma | representation.of_module_as_module_act | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"monoid_algebra",
"restrict_scalars",
"restrict_scalars.add_equiv",
"restrict_scalars.lsmul_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_of_module_as_module (r : monoid_algebra k G)
(m : (of_module k G M).as_module) :
(restrict_scalars.add_equiv _ _ _) ((of_module k G M).as_module_equiv (r • m)) =
r • (restrict_scalars.add_equiv _ _ _) ((of_module k G M).as_module_equiv m) | by { dsimp, simp only [add_equiv.apply_symm_apply, of_module_as_algebra_hom_apply_apply], } | lemma | representation.smul_of_module_as_module | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"monoid_algebra",
"restrict_scalars.add_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_action : representation k G (H →₀ k) | { to_fun := λ g, finsupp.lmap_domain k k ((•) g),
map_one' := by { ext x y, dsimp, simp },
map_mul' := λ x y, by { ext z w, simp [mul_smul] }} | def | representation.of_mul_action | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"finsupp.lmap_domain",
"representation"
] | A `G`-action on `H` induces a representation `G →* End(k[H])` in the natural way. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_mul_action_def (g : G) : of_mul_action k G H g = finsupp.lmap_domain k k ((•) g) | rfl | lemma | representation.of_mul_action_def | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"finsupp.lmap_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_action_single (g : G) (x : H) (r : k) :
of_mul_action k G H g (finsupp.single x r) = finsupp.single (g • x) r | finsupp.map_domain_single | lemma | representation.of_mul_action_single | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"finsupp.map_domain_single",
"finsupp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_action_apply {H : Type*} [mul_action G H]
(g : G) (f : H →₀ k) (h : H) : of_mul_action k G H g f h = f (g⁻¹ • h) | begin
conv_lhs { rw ← smul_inv_smul g h, },
let h' := g⁻¹ • h,
change of_mul_action k G H g f (g • h') = f h',
have hg : function.injective ((•) g : H → H), { intros h₁ h₂, simp, },
simp only [of_mul_action_def, finsupp.lmap_domain_apply, finsupp.map_domain_apply, hg],
end | lemma | representation.of_mul_action_apply | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"finsupp.lmap_domain_apply",
"finsupp.map_domain_apply",
"mul_action",
"smul_inv_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_action_self_smul_eq_mul
(x : monoid_algebra k G) (y : (of_mul_action k G G).as_module) :
x • y = (x * y : monoid_algebra k G) | x.induction_on (λ g, by show as_algebra_hom _ _ _ = _; ext; simp)
(λ x y hx hy, by simp only [hx, hy, add_mul, add_smul])
(λ r x hx, by show as_algebra_hom _ _ _ = _; simpa [←hx]) | lemma | representation.of_mul_action_self_smul_eq_mul | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"add_smul",
"monoid_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_action_self_as_module_equiv :
(of_mul_action k G G).as_module ≃ₗ[monoid_algebra k G] monoid_algebra k G | { map_smul' := of_mul_action_self_smul_eq_mul, ..as_module_equiv _ } | def | representation.of_mul_action_self_as_module_equiv | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"monoid_algebra"
] | If we equip `k[G]` with the `k`-linear `G`-representation induced by the left regular action of
`G` on itself, the resulting object is isomorphic as a `k[G]`-module to `k[G]` with its natural
`k[G]`-module structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_group_hom : G →* units (V →ₗ[k] V) | monoid_hom.to_hom_units ρ | def | representation.as_group_hom | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"monoid_hom.to_hom_units",
"units"
] | When `G` is a group, a `k`-linear representation of `G` on `V` can be thought of as
a group homomorphism from `G` into the invertible `k`-linear endomorphisms of `V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_group_hom_apply (g : G) : ↑(as_group_hom ρ g) = ρ g | by simp only [as_group_hom, monoid_hom.coe_to_hom_units] | lemma | representation.as_group_hom_apply | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"monoid_hom.coe_to_hom_units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tprod : representation k G (V ⊗[k] W) | { to_fun := λ g, tensor_product.map (ρV g) (ρW g),
map_one' := by simp only [map_one, tensor_product.map_one],
map_mul' := λ g h, by simp only [map_mul, tensor_product.map_mul] } | def | representation.tprod | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"map_mul",
"map_one",
"representation",
"tensor_product.map",
"tensor_product.map_mul",
"tensor_product.map_one"
] | Given representations of `G` on `V` and `W`, there is a natural representation of `G` on their
tensor product `V ⊗[k] W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tprod_apply (g : G) : (ρV ⊗ ρW) g = tensor_product.map (ρV g) (ρW g) | rfl | lemma | representation.tprod_apply | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"tensor_product.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_tprod_one_as_module (r : monoid_algebra k G) (x : V) (y : W) :
(r • (x ⊗ₜ y) : (ρV.tprod 1).as_module) = (r • x : ρV.as_module) ⊗ₜ y | begin
show as_algebra_hom _ _ _ = as_algebra_hom _ _ _ ⊗ₜ _,
simp only [as_algebra_hom_def, monoid_algebra.lift_apply,
tprod_apply, monoid_hom.one_apply, linear_map.finsupp_sum_apply,
linear_map.smul_apply, tensor_product.map_tmul, linear_map.one_apply],
simp only [finsupp.sum, tensor_product.sum_tmul],
... | lemma | representation.smul_tprod_one_as_module | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"linear_map.finsupp_sum_apply",
"linear_map.one_apply",
"linear_map.smul_apply",
"monoid_algebra",
"monoid_algebra.lift_apply",
"monoid_hom.one_apply",
"tensor_product.map_tmul",
"tensor_product.sum_tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_one_tprod_as_module (r : monoid_algebra k G) (x : V) (y : W) :
(r • (x ⊗ₜ y) : ((1 : representation k G V).tprod ρW).as_module) = x ⊗ₜ (r • y : ρW.as_module) | begin
show as_algebra_hom _ _ _ = _ ⊗ₜ as_algebra_hom _ _ _,
simp only [as_algebra_hom_def, monoid_algebra.lift_apply,
tprod_apply, monoid_hom.one_apply, linear_map.finsupp_sum_apply,
linear_map.smul_apply, tensor_product.map_tmul, linear_map.one_apply],
simp only [finsupp.sum, tensor_product.tmul_sum, te... | lemma | representation.smul_one_tprod_as_module | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"linear_map.finsupp_sum_apply",
"linear_map.one_apply",
"linear_map.smul_apply",
"monoid_algebra",
"monoid_algebra.lift_apply",
"monoid_hom.one_apply",
"representation",
"tensor_product.map_tmul",
"tensor_product.tmul_smul",
"tensor_product.tmul_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lin_hom : representation k G (V →ₗ[k] W) | { to_fun := λ g,
{ to_fun := λ f, (ρW g) ∘ₗ f ∘ₗ (ρV g⁻¹),
map_add' := λ f₁ f₂, by simp_rw [add_comp, comp_add],
map_smul' := λ r f, by simp_rw [ring_hom.id_apply, smul_comp, comp_smul]},
map_one' := linear_map.ext $ λ x,
by simp_rw [coe_mk, inv_one, map_one, one_apply, one_eq_id, comp_id, id_comp],
m... | def | representation.lin_hom | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"function.comp_apply",
"inv_one",
"linear_map.ext",
"map_mul",
"map_one",
"mul_inv_rev",
"representation",
"ring_hom.id_apply"
] | Given representations of `G` on `V` and `W`, there is a natural representation of `G` on the
module `V →ₗ[k] W`, where `G` acts by conjugation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lin_hom_apply (g : G) (f : V →ₗ[k] W) : (lin_hom ρV ρW) g f = (ρW g) ∘ₗ f ∘ₗ (ρV g⁻¹) | rfl | lemma | representation.lin_hom_apply | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : representation k G (module.dual k V) | { to_fun := λ g,
{ to_fun := λ f, f ∘ₗ (ρV g⁻¹),
map_add' := λ f₁ f₂, by simp only [add_comp],
map_smul' := λ r f,
by {ext, simp only [coe_comp, function.comp_app, smul_apply, ring_hom.id_apply]} },
map_one' :=
by {ext, simp only [coe_comp, function.comp_app, map_one, inv_one, coe_mk, one_apply]},... | def | representation.dual | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"inv_one",
"map_mul",
"map_one",
"module.dual",
"mul_inv_rev",
"representation",
"ring_hom.id_apply"
] | The dual of a representation `ρ` of `G` on a module `V`, given by `(dual ρ) g f = f ∘ₗ (ρ g⁻¹)`,
where `f : module.dual k V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_apply (g : G) : (dual ρV) g = module.dual.transpose (ρV g⁻¹) | rfl | lemma | representation.dual_apply | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"module.dual.transpose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_tensor_hom_comm (g : G) :
(dual_tensor_hom k V W) ∘ₗ (tensor_product.map (ρV.dual g) (ρW g)) =
(lin_hom ρV ρW) g ∘ₗ (dual_tensor_hom k V W) | begin
ext, simp [module.dual.transpose_apply],
end | lemma | representation.dual_tensor_hom_comm | representation_theory | src/representation_theory/basic.lean | [
"algebra.module.basic",
"algebra.module.linear_map",
"algebra.monoid_algebra.basic",
"linear_algebra.dual",
"linear_algebra.contraction",
"ring_theory.tensor_product"
] | [
"dual_tensor_hom",
"module.dual.transpose_apply",
"tensor_product.map"
] | Given $k$-modules $V, W$, there is a homomorphism $φ : V^* ⊗ W → Hom_k(V, W)$
(implemented by `linear_algebra.contraction.dual_tensor_hom`).
Given representations of $G$ on $V$ and $W$,there are representations of $G$ on $V^* ⊗ W$ and on
$Hom_k(V, W)$.
This lemma says that $φ$ is $G$-linear. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
character (V : fdRep k G) (g : G) | linear_map.trace k V (V.ρ g) | def | fdRep.character | representation_theory | src/representation_theory/character.lean | [
"representation_theory.fdRep",
"linear_algebra.trace",
"representation_theory.invariants"
] | [
"fdRep",
"linear_map.trace"
] | The character of a representation `V : fdRep k G` is the function associating to `g : G` the
trace of the linear map `V.ρ g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_mul_comm (V : fdRep k G) (g : G) (h : G) : V.character (h * g) = V.character (g * h) | by simp only [trace_mul_comm, character, map_mul] | lemma | fdRep.char_mul_comm | representation_theory | src/representation_theory/character.lean | [
"representation_theory.fdRep",
"linear_algebra.trace",
"representation_theory.invariants"
] | [
"fdRep",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_one (V : fdRep k G) : V.character 1 = finite_dimensional.finrank k V | by simp only [character, map_one, trace_one] | lemma | fdRep.char_one | representation_theory | src/representation_theory/character.lean | [
"representation_theory.fdRep",
"linear_algebra.trace",
"representation_theory.invariants"
] | [
"fdRep",
"finite_dimensional.finrank",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_tensor (V W : fdRep k G) : (V ⊗ W).character = V.character * W.character | by { ext g, convert trace_tensor_product' (V.ρ g) (W.ρ g) } | lemma | fdRep.char_tensor | representation_theory | src/representation_theory/character.lean | [
"representation_theory.fdRep",
"linear_algebra.trace",
"representation_theory.invariants"
] | [
"fdRep"
] | The character is multiplicative under the tensor product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_iso {V W : fdRep k G} (i : V ≅ W) : V.character = W.character | by { ext g, simp only [character, fdRep.iso.conj_ρ i], exact (trace_conj' (V.ρ g) _).symm } | lemma | fdRep.char_iso | representation_theory | src/representation_theory/character.lean | [
"representation_theory.fdRep",
"linear_algebra.trace",
"representation_theory.invariants"
] | [
"fdRep",
"fdRep.iso.conj_ρ"
] | The character of isomorphic representations is the same. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_conj (V : fdRep k G) (g : G) (h : G) :
V.character (h * g * h⁻¹) = V.character g | by rw [char_mul_comm, inv_mul_cancel_left] | lemma | fdRep.char_conj | representation_theory | src/representation_theory/character.lean | [
"representation_theory.fdRep",
"linear_algebra.trace",
"representation_theory.invariants"
] | [
"fdRep",
"inv_mul_cancel_left"
] | The character of a representation is constant on conjugacy classes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_dual (V : fdRep k G) (g : G) : (of (dual V.ρ)).character g = V.character g⁻¹ | trace_transpose' (V.ρ g⁻¹) | lemma | fdRep.char_dual | representation_theory | src/representation_theory/character.lean | [
"representation_theory.fdRep",
"linear_algebra.trace",
"representation_theory.invariants"
] | [
"fdRep"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_lin_hom (V W : fdRep k G) (g : G) :
(of (lin_hom V.ρ W.ρ)).character g = (V.character g⁻¹) * (W.character g) | by rw [←char_iso (dual_tensor_iso_lin_hom _ _), char_tensor, pi.mul_apply, char_dual] | lemma | fdRep.char_lin_hom | representation_theory | src/representation_theory/character.lean | [
"representation_theory.fdRep",
"linear_algebra.trace",
"representation_theory.invariants"
] | [
"fdRep",
"pi.mul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
average_char_eq_finrank_invariants (V : fdRep k G) :
⅟(fintype.card G : k) • ∑ g : G, V.character g = finrank k (invariants V.ρ) | by { rw ←(is_proj_average_map V.ρ).trace, simp [character, group_algebra.average, _root_.map_sum], } | theorem | fdRep.average_char_eq_finrank_invariants | representation_theory | src/representation_theory/character.lean | [
"representation_theory.fdRep",
"linear_algebra.trace",
"representation_theory.invariants"
] | [
"fdRep",
"fintype.card",
"group_algebra.average"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_orthonormal (V W : fdRep k G) [simple V] [simple W] :
⅟(fintype.card G : k) • ∑ g : G, V.character g * W.character g⁻¹ =
if nonempty (V ≅ W) then ↑1 else ↑0 | begin
-- First, we can rewrite the summand `V.character g * W.character g⁻¹` as the character
-- of the representation `V ⊗ W* ≅ Hom(W, V)` applied to `g`.
conv in (V.character _ * W.character _)
{ rw [mul_comm, ←char_dual, ←pi.mul_apply, ←char_tensor],
rw [char_iso (fdRep.dual_tensor_iso_lin_hom W.ρ V)], }... | lemma | fdRep.char_orthonormal | representation_theory | src/representation_theory/character.lean | [
"representation_theory.fdRep",
"linear_algebra.trace",
"representation_theory.invariants"
] | [
"fdRep",
"fdRep.dual_tensor_iso_lin_hom",
"fintype.card",
"mul_comm"
] | Orthogonality of characters for irreducible representations of finite group over an
algebraically closed field whose characteristic doesn't divide the order of the group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fdRep (k G : Type u) [field k] [monoid G] | Action (fgModule.{u} k) (Mon.of G) | abbreviation | fdRep | representation_theory | src/representation_theory/fdRep.lean | [
"representation_theory.Rep",
"algebra.category.fgModule.limits",
"category_theory.preadditive.schur",
"representation_theory.basic"
] | [
"Action",
"Mon.of",
"field",
"monoid"
] | The category of finite dimensional `k`-linear representations of a monoid `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ρ (V : fdRep k G) : G →* (V →ₗ[k] V) | V.ρ | def | fdRep.ρ | representation_theory | src/representation_theory/fdRep.lean | [
"representation_theory.Rep",
"algebra.category.fgModule.limits",
"category_theory.preadditive.schur",
"representation_theory.basic"
] | [
"fdRep"
] | The monoid homomorphism corresponding to the action of `G` onto `V : fdRep k G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_to_linear_equiv {V W : fdRep k G} (i : V ≅ W) : V ≃ₗ[k] W | fgModule.iso_to_linear_equiv ((Action.forget (fgModule k) (Mon.of G)).map_iso i) | def | fdRep.iso_to_linear_equiv | representation_theory | src/representation_theory/fdRep.lean | [
"representation_theory.Rep",
"algebra.category.fgModule.limits",
"category_theory.preadditive.schur",
"representation_theory.basic"
] | [
"Action.forget",
"Mon.of",
"fdRep",
"fgModule",
"fgModule.iso_to_linear_equiv"
] | The underlying `linear_equiv` of an isomorphism of representations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso.conj_ρ {V W : fdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (fdRep.iso_to_linear_equiv i).conj (V.ρ g) | begin
rw [fdRep.iso_to_linear_equiv, ←fgModule.iso.conj_eq_conj, iso.conj_apply],
rw [iso.eq_inv_comp ((Action.forget (fgModule k) (Mon.of G)).map_iso i)],
exact (i.hom.comm g).symm,
end | lemma | fdRep.iso.conj_ρ | representation_theory | src/representation_theory/fdRep.lean | [
"representation_theory.Rep",
"algebra.category.fgModule.limits",
"category_theory.preadditive.schur",
"representation_theory.basic"
] | [
"Action.forget",
"Mon.of",
"fdRep",
"fdRep.iso_to_linear_equiv",
"fgModule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of {V : Type u} [add_comm_group V] [module k V] [finite_dimensional k V]
(ρ : representation k G V) : fdRep k G | ⟨fgModule.of k V, ρ⟩ | def | fdRep.of | representation_theory | src/representation_theory/fdRep.lean | [
"representation_theory.Rep",
"algebra.category.fgModule.limits",
"category_theory.preadditive.schur",
"representation_theory.basic"
] | [
"add_comm_group",
"fdRep",
"finite_dimensional",
"module",
"representation"
] | Lift an unbundled representation to `fdRep`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_ρ (V : fdRep k G) : ((forget₂ (fdRep k G) (Rep k G)).obj V).ρ = V.ρ | by { ext g v, refl } | lemma | fdRep.forget₂_ρ | representation_theory | src/representation_theory/fdRep.lean | [
"representation_theory.Rep",
"algebra.category.fgModule.limits",
"category_theory.preadditive.schur",
"representation_theory.basic"
] | [
"Rep",
"fdRep"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finrank_hom_simple_simple [is_alg_closed k] (V W : fdRep k G) [simple V] [simple W] :
finrank k (V ⟶ W) = if nonempty (V ≅ W) then 1 else 0 | category_theory.finrank_hom_simple_simple k V W | lemma | fdRep.finrank_hom_simple_simple | representation_theory | src/representation_theory/fdRep.lean | [
"representation_theory.Rep",
"algebra.category.fgModule.limits",
"category_theory.preadditive.schur",
"representation_theory.basic"
] | [
"category_theory.finrank_hom_simple_simple",
"fdRep",
"is_alg_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_hom_linear_equiv (X Y : fdRep k G) :
(((forget₂ (fdRep k G) (Rep k G)).obj X) ⟶ ((forget₂ (fdRep k G) (Rep k G)).obj Y)) ≃ₗ[k]
(X ⟶ Y) | { to_fun := λ f, ⟨f.hom, f.comm⟩,
map_add' := λ _ _, rfl,
map_smul' := λ _ _, rfl,
inv_fun := λ f, ⟨(forget₂ (fgModule k) (Module k)).map f.hom, f.comm⟩,
left_inv := λ _, by { ext, refl },
right_inv := λ _, by { ext, refl } } | def | fdRep.forget₂_hom_linear_equiv | representation_theory | src/representation_theory/fdRep.lean | [
"representation_theory.Rep",
"algebra.category.fgModule.limits",
"category_theory.preadditive.schur",
"representation_theory.basic"
] | [
"Module",
"Rep",
"fdRep",
"fgModule",
"inv_fun"
] | The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_tensor_iso_lin_hom_aux :
((fdRep.of ρV.dual) ⊗ W).V ≅ (fdRep.of (lin_hom ρV W.ρ)).V | (dual_tensor_hom_equiv k V W).to_fgModule_iso | def | fdRep.dual_tensor_iso_lin_hom_aux | representation_theory | src/representation_theory/fdRep.lean | [
"representation_theory.Rep",
"algebra.category.fgModule.limits",
"category_theory.preadditive.schur",
"representation_theory.basic"
] | [
"dual_tensor_hom_equiv",
"fdRep.of"
] | Auxiliary definition for `fdRep.dual_tensor_iso_lin_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_tensor_iso_lin_hom :
(fdRep.of ρV.dual) ⊗ W ≅ fdRep.of (lin_hom ρV W.ρ) | begin
apply Action.mk_iso (dual_tensor_iso_lin_hom_aux ρV W),
convert (dual_tensor_hom_comm ρV W.ρ),
end | def | fdRep.dual_tensor_iso_lin_hom | representation_theory | src/representation_theory/fdRep.lean | [
"representation_theory.Rep",
"algebra.category.fgModule.limits",
"category_theory.preadditive.schur",
"representation_theory.basic"
] | [
"Action.mk_iso",
"fdRep.of"
] | When `V` and `W` are finite dimensional representations of a group `G`, the isomorphism
`dual_tensor_hom_equiv k V W` of vector spaces induces an isomorphism of representations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_tensor_iso_lin_hom_hom_hom :
(dual_tensor_iso_lin_hom ρV W).hom.hom = dual_tensor_hom k V W | rfl | lemma | fdRep.dual_tensor_iso_lin_hom_hom_hom | representation_theory | src/representation_theory/fdRep.lean | [
"representation_theory.Rep",
"algebra.category.fgModule.limits",
"category_theory.preadditive.schur",
"representation_theory.basic"
] | [
"dual_tensor_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
average : monoid_algebra k G | ⅟(fintype.card G : k) • ∑ g : G, of k G g | def | group_algebra.average | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [
"fintype.card",
"monoid_algebra"
] | The average of all elements of the group `G`, considered as an element of `monoid_algebra k G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_average_left (g : G) :
(finsupp.single g 1 * average k G : monoid_algebra k G) = average k G | begin
simp only [mul_one, finset.mul_sum, algebra.mul_smul_comm, average, monoid_algebra.of_apply,
finset.sum_congr, monoid_algebra.single_mul_single],
set f : G → monoid_algebra k G := λ x, finsupp.single x 1,
show ⅟ ↑(fintype.card G) • ∑ (x : G), f (g * x) = ⅟ ↑(fintype.card G) • ∑ (x : G), f x,
rw functi... | theorem | group_algebra.mul_average_left | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [
"algebra.mul_smul_comm",
"finset.mul_sum",
"finsupp.single",
"fintype.card",
"group.mul_left_bijective",
"monoid_algebra",
"monoid_algebra.single_mul_single",
"mul_one"
] | `average k G` is invariant under left multiplication by elements of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_average_right (g : G) :
average k G * finsupp.single g 1 = average k G | begin
simp only [mul_one, finset.sum_mul, algebra.smul_mul_assoc, average, monoid_algebra.of_apply,
finset.sum_congr, monoid_algebra.single_mul_single],
set f : G → monoid_algebra k G := λ x, finsupp.single x 1,
show ⅟ ↑(fintype.card G) • ∑ (x : G), f (x * g) = ⅟ ↑(fintype.card G) • ∑ (x : G), f x,
rw funct... | theorem | group_algebra.mul_average_right | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [
"algebra.smul_mul_assoc",
"finset.sum_mul",
"finsupp.single",
"fintype.card",
"group.mul_right_bijective",
"monoid_algebra",
"monoid_algebra.single_mul_single",
"mul_one"
] | `average k G` is invariant under right multiplication by elements of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
invariants : submodule k V | { carrier := set_of (λ v, ∀ (g : G), ρ g v = v),
zero_mem' := λ g, by simp only [map_zero],
add_mem' := λ v w hv hw g, by simp only [hv g, hw g, map_add],
smul_mem' := λ r v hv g, by simp only [hv g, linear_map.map_smulₛₗ, ring_hom.id_apply]} | def | representation.invariants | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [
"linear_map.map_smulₛₗ",
"ring_hom.id_apply",
"submodule"
] | The subspace of invariants, consisting of the vectors fixed by all elements of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_invariants (v : V) : v ∈ invariants ρ ↔ ∀ (g: G), ρ g v = v | by refl | lemma | representation.mem_invariants | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invariants_eq_inter :
(invariants ρ).carrier = ⋂ g : G, function.fixed_points (ρ g) | by {ext, simp [function.is_fixed_pt]} | lemma | representation.invariants_eq_inter | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [
"function.fixed_points",
"function.is_fixed_pt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
average_map : V →ₗ[k] V | as_algebra_hom ρ (average k G) | def | representation.average_map | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [] | The action of `average k G` gives a projection map onto the subspace of invariants. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
average_map_invariant (v : V) : average_map ρ v ∈ invariants ρ | λ g, by rw [average_map, ←as_algebra_hom_single_one, ←linear_map.mul_apply,
←map_mul (as_algebra_hom ρ), mul_average_left] | theorem | representation.average_map_invariant | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [] | The `average_map` sends elements of `V` to the subspace of invariants. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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