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 ⌀ |
|---|---|---|---|---|---|---|
res_obj_ρ {H : Type u} [Monoid H] (f : G →* H) (A : Rep k H) :
ρ ((Action.res _ f).obj A) = A.ρ.comp f := 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 | res_obj_ρ | null |
coe_res_obj_ρ {H : Type u} [Monoid H] (f : G →* H) (A : Rep k H) (g : G) :
DFunLike.coe (F := G →* (A →ₗ[k] A)) (ρ ((Action.res _ f).obj A)) g = A.ρ (f g) := rfl | 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 | coe_res_obj_ρ | null |
linearization : (Action (Type u) G) ⥤ (Rep k G) :=
(ModuleCat.free k).mapAction G | 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 | linearization | The monoidal functor sending a type `H` with a `G`-action to the induced `k`-linear
`G`-representation on `k[H].` |
@[simp]
coe_linearization_obj (X : Action (Type u) G) :
(linearization k G).obj X = (X.V →₀ k) := 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 | coe_linearization_obj | null |
linearization_obj_ρ (X : Action (Type u) G) (g : G) :
((linearization k G).obj X).ρ g = Finsupp.lmapDomain k k (X.ρ g) :=
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 | linearization_obj_ρ | null |
coe_linearization_obj_ρ (X : Action (Type u) G) (g : G) :
@DFunLike.coe (no_index G →* ((X.V →₀ k) →ₗ[k] (X.V →₀ k))) _
(fun _ => (X.V →₀ k) →ₗ[k] (X.V →₀ k)) _
((linearization k G).obj X).ρ g = Finsupp.lmapDomain k k (X.ρ g) := 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 | coe_linearization_obj_ρ | null |
linearization_single (X : Action (Type u) G) (g : G) (x : X.V) (r : k) :
((linearization k G).obj X).ρ g (Finsupp.single x r) = Finsupp.single (X.ρ g x) r := by
simp
@[deprecated "Use `Rep.linearization_single` instead" (since := "2025-06-02")] | 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 | linearization_single | null |
linearization_of (X : Action (Type u) G) (g : G) (x : X.V) :
((linearization k G).obj X).ρ g (Finsupp.single x (1 : k))
= Finsupp.single (X.ρ g x) (1 : k) := by
simp
variable {X Y : Action (Type u) G} (f : X ⟶ Y)
@[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 | linearization_of | null |
linearization_map_hom : ((linearization k G).map f).hom =
ModuleCat.ofHom (Finsupp.lmapDomain k k f.hom) :=
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 | linearization_map_hom | null |
linearization_map_hom_single (x : X.V) (r : k) :
((linearization k G).map f).hom (Finsupp.single x r) = Finsupp.single (f.hom x) r :=
Finsupp.mapDomain_single
open Functor.LaxMonoidal Functor.OplaxMonoidal Functor.Monoidal
@[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 | linearization_map_hom_single | null |
linearization_μ_hom (X Y : Action (Type u) G) :
(μ (linearization k G) X Y).hom =
ModuleCat.ofHom (finsuppTensorFinsupp' k X.V Y.V).toLinearMap :=
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 | linearization_μ_hom | null |
linearization_δ_hom (X Y : Action (Type u) G) :
(δ (linearization k G) X Y).hom =
ModuleCat.ofHom (finsuppTensorFinsupp' k X.V Y.V).symm.toLinearMap :=
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 | linearization_δ_hom | null |
linearization_ε_hom : (ε (linearization k G)).hom =
ModuleCat.ofHom (Finsupp.lsingle PUnit.unit) :=
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 | linearization_ε_hom | null |
linearization_η_hom_apply (r : k) :
(η (linearization k G)).hom (Finsupp.single PUnit.unit r) = r :=
(εIso (linearization k G)).hom_inv_id_apply r
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 | linearization_η_hom_apply | null |
@[simps! hom_hom inv_hom]
linearizationTrivialIso (X : Type u) :
(linearization k G).obj (Action.mk X 1) ≅ trivial k G (X →₀ k) :=
Action.mkIso (Iso.refl _) fun _ => ModuleCat.hom_ext <| Finsupp.lhom_ext' fun _ => LinearMap.ext
fun _ => linearization_single .. | 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 | linearizationTrivialIso | The linearization of a type `X` on which `G` acts trivially is the trivial `G`-representation
on `k[X]`. |
ofMulAction (H : Type u) [MulAction G H] : Rep k G :=
of <| Representation.ofMulAction k G H | 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 | ofMulAction | Given a `G`-action on `H`, this is `k[H]` bundled with the natural representation
`G →* End(k[H])` as a term of type `Rep k G`. |
leftRegular : Rep k G :=
ofMulAction k G 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 | leftRegular | The `k`-linear `G`-representation on `k[G]`, induced by left multiplication. |
diagonal (n : ℕ) : Rep k G :=
ofMulAction k G (Fin n → 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 | diagonal | The `k`-linear `G`-representation on `k[Gⁿ]`, induced by left multiplication. |
@[simps! hom_hom inv_hom]
diagonalOneIsoLeftRegular :
diagonal k G 1 ≅ leftRegular k G :=
Action.mkIso (Finsupp.domLCongr <| Equiv.funUnique (Fin 1) G).toModuleIso fun _ =>
ModuleCat.hom_ext <| Finsupp.lhom_ext fun _ _ => by simp [diagonal, ModuleCat.endRingEquiv] | 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 | diagonalOneIsoLeftRegular | The natural isomorphism between the representations on `k[G¹]` and `k[G]` induced by left
multiplication in `G`. |
@[simps! hom_hom inv_hom]
ofMulActionSubsingletonIsoTrivial
(H : Type u) [Subsingleton H] [MulOneClass H] [MulAction G H] :
ofMulAction k G H ≅ trivial k G k :=
letI : Unique H := uniqueOfSubsingleton 1
Action.mkIso (Finsupp.LinearEquiv.finsuppUnique _ _ _).toModuleIso fun _ =>
ModuleCat.hom_ext <| Fins... | 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 | ofMulActionSubsingletonIsoTrivial | When `H = {1}`, the `G`-representation on `k[H]` induced by an action of `G` on `H` is
isomorphic to the trivial representation on `k`. |
linearizationOfMulActionIso (H : Type u) [MulAction G H] :
(linearization k G).obj (Action.ofMulAction G H) ≅ ofMulAction k G H :=
Iso.refl _ | 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 | linearizationOfMulActionIso | The linearization of a type `H` with a `G`-action is definitionally isomorphic to the
`k`-linear `G`-representation on `k[H]` induced by the `G`-action on `H`. |
ofDistribMulAction : Rep k G := Rep.of (Representation.ofDistribMulAction k G A)
@[simp] theorem ofDistribMulAction_ρ_apply_apply (g : G) (a : A) :
(ofDistribMulAction k G A).ρ g a = g • a := 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 | ofDistribMulAction | Turns a `k`-module `A` with a compatible `DistribMulAction` of a monoid `G` into a
`k`-linear `G`-representation on `A`. |
@[simp] ofAlgebraAut (R S : Type) [CommRing R] [CommRing S] [Algebra R S] :
Rep ℤ (S ≃ₐ[R] S) := ofDistribMulAction ℤ (S ≃ₐ[R] S) S | 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 | ofAlgebraAut | Given an `R`-algebra `S`, the `ℤ`-linear representation associated to the natural action of
`S ≃ₐ[R] S` on `S`. |
ofMulDistribMulAction : Rep ℤ M := Rep.of (Representation.ofMulDistribMulAction M G)
@[simp] theorem ofMulDistribMulAction_ρ_apply_apply (g : M) (a : Additive G) :
(ofMulDistribMulAction M G).ρ g a = Additive.ofMul (g • a.toMul) := 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 | ofMulDistribMulAction | Turns a `CommGroup` `G` with a `MulDistribMulAction` of a monoid `M` into a
`ℤ`-linear `M`-representation on `Additive G`. |
@[simp] ofAlgebraAutOnUnits (R S : Type) [CommRing R] [CommRing S] [Algebra R S] :
Rep ℤ (S ≃ₐ[R] S) := Rep.ofMulDistribMulAction (S ≃ₐ[R] S) Sˣ | 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 | ofAlgebraAutOnUnits | Given an `R`-algebra `S`, the `ℤ`-linear representation associated to the natural action of
`S ≃ₐ[R] S` on `Sˣ`. |
@[simps]
leftRegularHom (A : Rep k G) (x : A) : leftRegular k G ⟶ A where
hom := ModuleCat.ofHom <| Finsupp.lift A k G fun g => A.ρ g x
comm _ := by ext; simp [ModuleCat.endRingEquiv] | 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 | leftRegularHom | Given an element `x : A`, there is a natural morphism of representations `k[G] ⟶ A` sending
`g ↦ A.ρ(g)(x).` |
leftRegularHom_hom_single {A : Rep k G} (g : G) (x : A) (r : k) :
(leftRegularHom A x).hom (Finsupp.single g r) = r • A.ρ g x := by 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 | leftRegularHom_hom_single | null |
@[simps]
leftRegularHomEquiv (A : Rep k G) : (leftRegular k G ⟶ A) ≃ₗ[k] A where
toFun f := f.hom (Finsupp.single 1 1)
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun x := leftRegularHom A x
left_inv f := by ext; simp [← hom_comm_apply f]
right_inv x := by simp | 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 | leftRegularHomEquiv | Given a `k`-linear `G`-representation `A`, there is a `k`-linear isomorphism between
representation morphisms `Hom(k[G], A)` and `A`. |
leftRegularHomEquiv_symm_single {A : Rep k G} (x : A) (g : G) :
((leftRegularHomEquiv A).symm x).hom (Finsupp.single g 1) = A.ρ g x := by
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 | leftRegularHomEquiv_symm_single | null |
finsupp : Rep k G :=
Rep.of (Representation.finsupp A.ρ α)
variable (k G) in | 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 | finsupp | The representation on `α →₀ A` defined pointwise by a representation on `A`. |
free : Rep k G :=
Rep.of (V := (α →₀ G →₀ k)) (Representation.free k G α)
variable {α} [DecidableEq α] | 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 | free | The representation on `α →₀ k[G]` defined pointwise by the left regular representation on
`k[G]`. |
@[simps]
freeLift (f : α → A) :
free k G α ⟶ A where
hom := ModuleCat.ofHom <| linearCombination k (fun x => A.ρ x.2 (f x.1)) ∘ₗ
(finsuppProdLEquiv k).symm.toLinearMap
comm _ := by
ext; simp [ModuleCat.endRingEquiv]
variable {A} in | 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 | freeLift | Given `f : α → A`, the natural representation morphism `(α →₀ k[G]) ⟶ A` sending
`single a (single g r) ↦ r • A.ρ g (f a)`. |
freeLift_hom_single_single (f : α → A) (i : α) (g : G) (r : k) :
(freeLift A f).hom (single i (single g r)) = r • A.ρ g (f i) := by
simp
variable (α) in | 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 | freeLift_hom_single_single | null |
@[simps]
freeLiftLEquiv :
(free k G α ⟶ A) ≃ₗ[k] (α → A) where
toFun f i := f.hom (single i (single 1 1))
invFun := freeLift A
left_inv x := by
ext i j
simpa [← map_smul] using (hom_comm_apply x j (single i (single 1 1))).symm
right_inv _ := by ext; simp
map_add' _ _ := rfl
map_smul' _ _ := ... | 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 | freeLiftLEquiv | The natural linear equivalence between functions `α → A` and representation morphisms
`(α →₀ k[G]) ⟶ A`. |
free_ext (f g : free k G α ⟶ A)
(h : ∀ i : α, f.hom (single i (single 1 1)) = g.hom (single i (single 1 1))) : f = g := by
classical exact (freeLiftLEquiv α A).injective (funext_iff.2 h) | 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 | free_ext | null |
@[simps! hom_hom inv_hom]
finsuppTensorLeft :
A.finsupp α ⊗ B ≅ (A ⊗ B).finsupp α :=
Action.mkIso (TensorProduct.finsuppLeft k A B α).toModuleIso
fun _ => ModuleCat.hom_ext <| TensorProduct.ext <| lhom_ext fun _ _ => by
ext
simp [Action_ρ_eq_ρ, TensorProduct.finsuppLeft_apply_tmul, tensorObj_def,
... | 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 | finsuppTensorLeft | Given representations `A, B` and a type `α`, this is the natural representation isomorphism
`(α →₀ A) ⊗ B ≅ (A ⊗ B) →₀ α` sending `single x a ⊗ₜ b ↦ single x (a ⊗ₜ b)`. |
@[simps! hom_hom inv_hom]
finsuppTensorRight :
A ⊗ B.finsupp α ≅ (A ⊗ B).finsupp α :=
Action.mkIso (TensorProduct.finsuppRight k A B α).toModuleIso fun _ => ModuleCat.hom_ext <|
TensorProduct.ext <| LinearMap.ext fun _ => lhom_ext fun _ _ => by
ext
simp [Action_ρ_eq_ρ, TensorProduct.finsuppRight... | 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 | finsuppTensorRight | Given representations `A, B` and a type `α`, this is the natural representation isomorphism
`A ⊗ (α →₀ B) ≅ (A ⊗ B) →₀ α` sending `a ⊗ₜ single x b ↦ single x (a ⊗ₜ b)`. |
@[simps! -isSimp hom_hom inv_hom]
leftRegularTensorTrivialIsoFree :
leftRegular k G ⊗ trivial k G (α →₀ k) ≅ free k G α :=
Action.mkIso (finsuppTensorFinsupp' k G α ≪≫ₗ Finsupp.domLCongr (Equiv.prodComm G α) ≪≫ₗ
finsuppProdLEquiv k).toModuleIso fun _ =>
ModuleCat.hom_ext <| TensorProduct.ext <| lhom_ext... | 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 | leftRegularTensorTrivialIsoFree | The natural isomorphism sending `single g r₁ ⊗ single a r₂ ↦ single a (single g r₁r₂)`. |
leftRegularTensorTrivialIsoFree_hom_hom_single_tmul_single (i : α) (g : G) (r s : k) :
DFunLike.coe (F := ↑(ModuleCat.of k (G →₀ k) ⊗ ModuleCat.of k (α →₀ k)) →ₗ[k] α →₀ G →₀ k)
(leftRegularTensorTrivialIsoFree k G α).hom.hom.hom (single g r ⊗ₜ[k] single i s) =
single i (single g (r * s)) := by
simp [le... | 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 | leftRegularTensorTrivialIsoFree_hom_hom_single_tmul_single | null |
leftRegularTensorTrivialIsoFree_inv_hom_single_single (i : α) (g : G) (r : k) :
DFunLike.coe (F := (α →₀ G →₀ k) →ₗ[k] ↑(ModuleCat.of k (G →₀ k) ⊗ ModuleCat.of k (α →₀ k)))
(leftRegularTensorTrivialIsoFree k G α).inv.hom.hom (single i (single g r)) =
single g r ⊗ₜ[k] single i 1 := by
simp [leftRegularTe... | 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 | leftRegularTensorTrivialIsoFree_inv_hom_single_single | null |
diagonalSuccIsoTensorTrivial :
diagonal k G (n + 1) ≅ leftRegular k G ⊗ trivial k G ((Fin n → G) →₀ k) :=
(linearization k G).mapIso (Action.diagonalSuccIsoTensorTrivial G n) ≪≫
(Functor.Monoidal.μIso (linearization k G) _ _).symm ≪≫
tensorIso (Iso.refl _) (linearizationTrivialIso k G (Fin n → G))
@[sim... | 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 | diagonalSuccIsoTensorTrivial | An isomorphism of `k`-linear representations of `G` from `k[Gⁿ⁺¹]` to `k[G] ⊗ₖ k[Gⁿ]` (on
which `G` acts by `ρ(g₁)(g₂ ⊗ x) = (g₁ * g₂) ⊗ x`) sending `(g₀, ..., gₙ)` to
`g₀ ⊗ (g₀⁻¹g₁, g₁⁻¹g₂, ..., gₙ₋₁⁻¹gₙ)`. The inverse sends `g₀ ⊗ (g₁, ..., gₙ)` to
`(g₀, g₀g₁, ..., g₀g₁...gₙ)`. |
diagonalSuccIsoTensorTrivial_hom_hom_single (f : Fin (n + 1) → G) (a : k) :
DFunLike.coe (F := ((Fin (n + 1) → G) →₀ k) →ₗ[k]
↑(ModuleCat.of k (G →₀ k) ⊗ ModuleCat.of k ((Fin n → G) →₀ k)))
(diagonalSuccIsoTensorTrivial k G n).hom.hom.hom (single f a) =
single (f 0) 1 ⊗ₜ single (fun i => (f (Fin.cas... | 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 | diagonalSuccIsoTensorTrivial_hom_hom_single | null |
diagonalSuccIsoTensorTrivial_inv_hom_single_single (g : G) (f : Fin n → G) (a b : k) :
(diagonalSuccIsoTensorTrivial k G n).inv.hom (single g a ⊗ₜ single f b) =
single (g • Fin.partialProd f) (a * b) := by
have := Action.diagonalSuccIsoTensorTrivial_inv_hom_apply (G := G) (n := n)
simp_all [diagonalSuccIs... | 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 | diagonalSuccIsoTensorTrivial_inv_hom_single_single | null |
diagonalSuccIsoTensorTrivial_inv_hom_single_left (g : G) (f : (Fin n → G) →₀ k) (r : k) :
(diagonalSuccIsoTensorTrivial k G n).inv.hom (single g r ⊗ₜ f) =
Finsupp.lift ((Fin (n + 1) → G) →₀ k) k (Fin n → G)
(fun f => single (g • Fin.partialProd f) r) f := by
refine f.induction ?_ ?_
· simp only [Ten... | 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 | diagonalSuccIsoTensorTrivial_inv_hom_single_left | null |
diagonalSuccIsoTensorTrivial_inv_hom_single_right (g : G →₀ k) (f : Fin n → G) (r : k) :
(diagonalSuccIsoTensorTrivial k G n).inv.hom (g ⊗ₜ single f r) =
Finsupp.lift _ k G (fun a => single (a • Fin.partialProd f) r) g := by
refine g.induction ?_ ?_
· simp only [TensorProduct.zero_tmul, map_zero]
· intr... | 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 | diagonalSuccIsoTensorTrivial_inv_hom_single_right | null |
diagonalSuccIsoFree : diagonal k G (n + 1) ≅ free k G (Fin n → G) :=
diagonalSuccIsoTensorTrivial k G n ≪≫ leftRegularTensorTrivialIsoFree k G (Fin n → G)
@[simp] | 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 | diagonalSuccIsoFree | Representation isomorphism `k[Gⁿ⁺¹] ≅ (Gⁿ →₀ k[G])`, where the right-hand representation is
defined pointwise by the left regular representation on `k[G]`. The map sends
`single (g₀, ..., gₙ) a ↦ single (g₀⁻¹g₁, ..., gₙ₋₁⁻¹gₙ) (single g₀ a)`. |
diagonalSuccIsoFree_hom_hom_single (f : Fin (n + 1) → G) (a : k) :
(diagonalSuccIsoFree k G n).hom.hom (single f a) =
single (fun i => (f i.castSucc)⁻¹ * f i.succ) (single (f 0) a) := by
simp [diagonalSuccIsoFree, leftRegularTensorTrivialIsoFree_hom_hom_single_tmul_single
(k := k)]
@[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 | diagonalSuccIsoFree_hom_hom_single | null |
diagonalSuccIsoFree_inv_hom_single_single (g : G) (f : Fin n → G) (a : k) :
(diagonalSuccIsoFree k G n).inv.hom (single f (single g a)) =
single (g • Fin.partialProd f) a := by
have := diagonalSuccIsoTensorTrivial_inv_hom_single_single g f a 1
simp_all [diagonalSuccIsoFree, leftRegularTensorTrivialIsoFree... | 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 | diagonalSuccIsoFree_inv_hom_single_single | null |
diagonalSuccIsoFree_inv_hom_single (g : G →₀ k) (f : Fin n → G) :
(diagonalSuccIsoFree k G n).inv.hom (single f g) =
lift _ k G (fun a => single (a • Fin.partialProd f) 1) g :=
g.induction (by simp) fun _ _ _ _ _ _ => by
rw [single_add, map_add, diagonalSuccIsoFree_inv_hom_single_single]
simp_all [s... | 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 | diagonalSuccIsoFree_inv_hom_single | null |
diagonalHomEquiv :
(Rep.diagonal k G (n + 1) ⟶ A) ≃ₗ[k] (Fin n → G) → A :=
Linear.homCongr k (diagonalSuccIsoFree k G n) (Iso.refl _) ≪≫ₗ
freeLiftLEquiv (Fin n → G) A
variable {n A} | 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 | diagonalHomEquiv | Given a `k`-linear `G`-representation `A`, the set of representation morphisms
`Hom(k[Gⁿ⁺¹], A)` is `k`-linearly isomorphic to the set of functions `Gⁿ → A`. |
diagonalHomEquiv_apply (f : Rep.diagonal k G (n + 1) ⟶ A) (x : Fin n → G) :
diagonalHomEquiv n A f x = f.hom (Finsupp.single (Fin.partialProd x) 1) := by
simp [diagonalHomEquiv, Linear.homCongr_apply,
diagonalSuccIsoFree_inv_hom_single_single (k := k)] | 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 | diagonalHomEquiv_apply | Given a `k`-linear `G`-representation `A`, `diagonalHomEquiv` is a `k`-linear isomorphism of
the set of representation morphisms `Hom(k[Gⁿ⁺¹], A)` with `Fun(Gⁿ, A)`. This lemma says that this
sends a morphism of representations `f : k[Gⁿ⁺¹] ⟶ A` to the function
`(g₁, ..., gₙ) ↦ f(1, g₁, g₁g₂, ..., g₁g₂...gₙ).` |
diagonalHomEquiv_symm_apply (f : (Fin n → G) → A) (x : Fin (n + 1) → G) :
((diagonalHomEquiv n A).symm f).hom (Finsupp.single x 1) =
A.ρ (x 0) (f fun i : Fin n => (x (Fin.castSucc i))⁻¹ * x i.succ) := by
simp [diagonalHomEquiv, Linear.homCongr_symm_apply, diagonalSuccIsoFree_hom_hom_single (k := k)] | 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 | diagonalHomEquiv_symm_apply | Given a `k`-linear `G`-representation `A`, `diagonalHomEquiv` is a `k`-linear isomorphism of
the set of representation morphisms `Hom(k[Gⁿ⁺¹], A)` with `Fun(Gⁿ, A)`. This lemma says that the
inverse map sends a function `f : Gⁿ → A` to the representation morphism sending
`(g₀, ... gₙ) ↦ ρ(g₀)(f(g₀⁻¹g₁, g₁⁻¹g₂, ..., gₙ₋... |
@[deprecated "We no longer use `diagonalHomEquiv` to define group cohomology"
(since := "2025-06-08")]
diagonalHomEquiv_symm_partialProd_succ (f : (Fin n → G) → A) (g : Fin (n + 1) → G)
(a : Fin (n + 1)) :
((diagonalHomEquiv n A).symm f).hom (Finsupp.single (Fin.partialProd g ∘ a.succ.succAbove) 1)
= f (F... | 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 | diagonalHomEquiv_symm_partialProd_succ | Auxiliary lemma for defining group cohomology, used to show that the isomorphism
`diagonalHomEquiv` commutes with the differentials in two complexes which compute
group cohomology. |
@[simps]
norm : End A where
hom := ModuleCat.ofHom <| Representation.norm A.ρ
comm g := by ext; simp
@[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 | norm | Given a representation `A` of a finite group `G`, `norm A` is the representation morphism
`A ⟶ A` defined by `x ↦ ∑ A.ρ g x` for `g` in `G`. |
norm_comm {A B : Rep k G} (f : A ⟶ B) : f ≫ norm B = norm A ≫ f := by
ext
simp [Representation.norm, 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 | norm_comm | null |
@[simps]
normNatTrans : End (𝟭 (Rep k G)) where
app := norm
naturality _ _ := norm_comm | 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 | normNatTrans | Given a representation `A` of a finite group `G`, the norm map `A ⟶ A` defined by
`x ↦ ∑ A.ρ g x` for `g` in `G` defines a natural endomorphism of the identity functor. |
@[simps]
protected noncomputable ihom (A : Rep k G) : Rep k G ⥤ Rep k G where
obj B := Rep.of (Representation.linHom A.ρ B.ρ)
map := fun {X} {Y} f =>
{ hom := ModuleCat.ofHom (LinearMap.llcomp k _ _ _ f.hom.hom)
comm g := by ext; simp [ModuleCat.endRingEquiv, hom_comm_apply] }
map_id := fun _ => by ext;... | 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 | ihom | Given a `k`-linear `G`-representation `(A, ρ₁)`, this is the 'internal Hom' functor sending
`(B, ρ₂)` to the representation `Homₖ(A, B)` that maps `g : G` and `f : A →ₗ[k] B` to
`(ρ₂ g) ∘ₗ f ∘ₗ (ρ₁ g⁻¹)`. |
@[simps]
homEquiv (A B C : Rep k G) : (A ⊗ B ⟶ C) ≃ (B ⟶ (Rep.ihom A).obj C) where
toFun f :=
{ hom := ModuleCat.ofHom <| (TensorProduct.curry f.hom.hom).flip
comm g := ModuleCat.hom_ext <| LinearMap.ext fun x => LinearMap.ext fun y => by
simpa [ModuleCat.MonoidalCategory.tensorObj_def,
Mo... | 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 | homEquiv | Given a `k`-linear `G`-representation `A`, this is the Hom-set bijection in the adjunction
`A ⊗ - ⊣ ihom(A, -)`. It sends `f : A ⊗ B ⟶ C` to a `Rep k G` morphism defined by currying the
`k`-linear map underlying `f`, giving a map `A →ₗ[k] B →ₗ[k] C`, then flipping the arguments. |
@[simp]
ihom_obj_ρ_def (A B : Rep k G) : ((ihom A).obj B).ρ = ((Rep.ihom A).obj 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 | ihom_obj_ρ_def | null |
homEquiv_def (A B C : Rep k G) : (ihom.adjunction A).homEquiv B C = Rep.homEquiv A B C :=
congrFun (congrFun (Adjunction.mkOfHomEquiv_homEquiv _) _) _
@[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 | homEquiv_def | null |
ihom_ev_app_hom (A B : Rep k G) :
Action.Hom.hom ((ihom.ev A).app B) = ModuleCat.ofHom
(TensorProduct.uncurry k A (A →ₗ[k] B) B LinearMap.id.flip) := by
ext; rfl
@[simp] theorem ihom_coev_app_hom (A B : Rep k G) :
Action.Hom.hom ((ihom.coev A).app B) = ModuleCat.ofHom (TensorProduct.mk k _ _).flip :=
... | 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 | ihom_ev_app_hom | null |
MonoidalClosed.linearHomEquiv : (A ⊗ B ⟶ C) ≃ₗ[k] B ⟶ A ⟶[Rep k G] C :=
{ (ihom.adjunction A).homEquiv _ _ with
map_add' := fun _ _ => rfl
map_smul' := 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 | MonoidalClosed.linearHomEquiv | There is a `k`-linear isomorphism between the sets of representation morphisms`Hom(A ⊗ B, C)`
and `Hom(B, Homₖ(A, C))`. |
MonoidalClosed.linearHomEquivComm : (A ⊗ B ⟶ C) ≃ₗ[k] A ⟶ B ⟶[Rep k G] C :=
Linear.homCongr k (β_ A B) (Iso.refl _) ≪≫ₗ MonoidalClosed.linearHomEquiv _ _ _
variable {A B C}
@[simp] | 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 | MonoidalClosed.linearHomEquivComm | There is a `k`-linear isomorphism between the sets of representation morphisms`Hom(A ⊗ B, C)`
and `Hom(A, Homₖ(B, C))`. |
MonoidalClosed.linearHomEquiv_hom (f : A ⊗ B ⟶ C) :
(MonoidalClosed.linearHomEquiv A B C f).hom =
ModuleCat.ofHom (TensorProduct.curry f.hom.hom).flip :=
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 | MonoidalClosed.linearHomEquiv_hom | null |
MonoidalClosed.linearHomEquivComm_hom (f : A ⊗ B ⟶ C) :
(MonoidalClosed.linearHomEquivComm A B C f).hom =
ModuleCat.ofHom (TensorProduct.curry f.hom.hom) :=
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 | MonoidalClosed.linearHomEquivComm_hom | null |
MonoidalClosed.linearHomEquiv_symm_hom (f : B ⟶ A ⟶[Rep k G] C) :
((MonoidalClosed.linearHomEquiv A B C).symm f).hom =
ModuleCat.ofHom (TensorProduct.uncurry k A B C f.hom.hom.flip) := by
simp [linearHomEquiv]
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 | MonoidalClosed.linearHomEquiv_symm_hom | null |
MonoidalClosed.linearHomEquivComm_symm_hom (f : A ⟶ B ⟶[Rep k G] C) :
((MonoidalClosed.linearHomEquivComm A B C).symm f).hom =
ModuleCat.ofHom (TensorProduct.uncurry k A B C f.hom.hom) :=
ModuleCat.hom_ext <| TensorProduct.ext' fun _ _ => 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 | MonoidalClosed.linearHomEquivComm_symm_hom | null |
repOfTprodIso : Rep.of (ρ.tprod τ) ≅ Rep.of ρ ⊗ Rep.of τ :=
Iso.refl _ | 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 | repOfTprodIso | Tautological isomorphism to help Lean in typechecking. |
repOfTprodIso_apply (x : TensorProduct k V W) : (repOfTprodIso ρ τ).hom.hom x = x :=
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 | repOfTprodIso_apply | null |
repOfTprodIso_inv_apply (x : TensorProduct k V W) : (repOfTprodIso ρ τ).inv.hom x = x :=
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 | repOfTprodIso_inv_apply | null |
to_Module_monoidAlgebra_map_aux {k G : Type*} [CommRing k] [Monoid G] (V W : Type*)
[AddCommGroup V] [AddCommGroup W] [Module k V] [Module k W] (ρ : G →* V →ₗ[k] V)
(σ : G →* W →ₗ[k] W) (f : V →ₗ[k] W) (w : ∀ g : G, f.comp (ρ g) = (σ g).comp f)
(r : MonoidAlgebra k G) (x : V) :
f ((((MonoidAlgebra.lift ... | 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 | to_Module_monoidAlgebra_map_aux | Auxiliary lemma for `toModuleMonoidAlgebra`. |
toModuleMonoidAlgebraMap {V W : Rep k G} (f : V ⟶ W) :
ModuleCat.of (MonoidAlgebra k G) V.ρ.asModule ⟶ ModuleCat.of (MonoidAlgebra k G) W.ρ.asModule :=
ModuleCat.ofHom
{ f.hom.hom with
map_smul' := fun r x => to_Module_monoidAlgebra_map_aux V.V W.V V.ρ W.ρ f.hom.hom
(fun g => ModuleCat.hom_ext_i... | 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 | toModuleMonoidAlgebraMap | Auxiliary definition for `toModuleMonoidAlgebra`. |
toModuleMonoidAlgebra : Rep k G ⥤ ModuleCat.{u} (MonoidAlgebra k G) where
obj V := ModuleCat.of _ V.ρ.asModule
map f := toModuleMonoidAlgebraMap f | 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 | toModuleMonoidAlgebra | Functorially convert a representation of `G` into a module over `MonoidAlgebra k G`. |
ofModuleMonoidAlgebra : ModuleCat.{u} (MonoidAlgebra k G) ⥤ Rep k G where
obj M := Rep.of (Representation.ofModule M)
map f :=
{ hom := ModuleCat.ofHom
{ f.hom with
map_smul' := fun r x => f.hom.map_smul (algebraMap k _ r) x }
comm := fun g => by ext; apply f.hom.map_smul } | 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 | ofModuleMonoidAlgebra | Functorially convert a module over `MonoidAlgebra k G` into a representation of `G`. |
ofModuleMonoidAlgebra_obj_coe (M : ModuleCat.{u} (MonoidAlgebra k G)) :
(ofModuleMonoidAlgebra.obj M : Type u) = RestrictScalars k (MonoidAlgebra k G) M :=
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 | ofModuleMonoidAlgebra_obj_coe | null |
ofModuleMonoidAlgebra_obj_ρ (M : ModuleCat.{u} (MonoidAlgebra k G)) :
(ofModuleMonoidAlgebra.obj M).ρ = Representation.ofModule M :=
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 | ofModuleMonoidAlgebra_obj_ρ | null |
counitIsoAddEquiv {M : ModuleCat.{u} (MonoidAlgebra k G)} :
(ofModuleMonoidAlgebra ⋙ toModuleMonoidAlgebra).obj M ≃+ M := by
dsimp [ofModuleMonoidAlgebra, toModuleMonoidAlgebra]
exact (Representation.ofModule M).asModuleEquiv.toAddEquiv.trans
(RestrictScalars.addEquiv k (MonoidAlgebra k G) _) | 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 | counitIsoAddEquiv | Auxiliary definition for `equivalenceModuleMonoidAlgebra`. |
unitIsoAddEquiv {V : Rep k G} : V ≃+ (toModuleMonoidAlgebra ⋙ ofModuleMonoidAlgebra).obj V := by
dsimp [ofModuleMonoidAlgebra, toModuleMonoidAlgebra]
exact V.ρ.asModuleEquiv.symm.toAddEquiv.trans (RestrictScalars.addEquiv _ _ _).symm | 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 | unitIsoAddEquiv | Auxiliary definition for `equivalenceModuleMonoidAlgebra`. |
counitIso (M : ModuleCat.{u} (MonoidAlgebra k G)) :
(ofModuleMonoidAlgebra ⋙ toModuleMonoidAlgebra).obj M ≅ M :=
LinearEquiv.toModuleIso
{ counitIsoAddEquiv with
map_smul' := fun r x => by
dsimp [counitIsoAddEquiv]
simp } | 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 | counitIso | Auxiliary definition for `equivalenceModuleMonoidAlgebra`. |
unit_iso_comm (V : Rep k G) (g : G) (x : V) :
unitIsoAddEquiv ((V.ρ g).toFun x) = ((ofModuleMonoidAlgebra.obj
(toModuleMonoidAlgebra.obj V)).ρ g).toFun (unitIsoAddEquiv x) := by
simp [unitIsoAddEquiv, ofModuleMonoidAlgebra, toModuleMonoidAlgebra] | 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 | unit_iso_comm | null |
unitIso (V : Rep k G) : V ≅ (toModuleMonoidAlgebra ⋙ ofModuleMonoidAlgebra).obj V :=
Action.mkIso
(LinearEquiv.toModuleIso
{ unitIsoAddEquiv with
map_smul' := fun r x => by
change (RestrictScalars.addEquiv _ _ _).symm (V.ρ.asModuleEquiv.symm (r • x)) = _
simp only [Representation... | 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 | unitIso | Auxiliary definition for `equivalenceModuleMonoidAlgebra`. |
equivalenceModuleMonoidAlgebra : Rep k G ≌ ModuleCat.{u} (MonoidAlgebra k G) where
functor := toModuleMonoidAlgebra
inverse := ofModuleMonoidAlgebra
unitIso := NatIso.ofComponents (fun V => unitIso V) (by cat_disch)
counitIso := NatIso.ofComponents (fun M => counitIso M) (by cat_disch) | 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 | equivalenceModuleMonoidAlgebra | The categorical equivalence `Rep k G ≌ ModuleCat (MonoidAlgebra k G)`. |
free_projective {G α : Type u} [Group G] :
Projective (free k G α) :=
equivalenceModuleMonoidAlgebra.toAdjunction.projective_of_map_projective _ <|
@ModuleCat.projective_of_free.{u} _ _
(ModuleCat.of (MonoidAlgebra k G) (Representation.free k G α).asModule)
_ (Representation.freeAsModuleBasis k G ... | instance | 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 | free_projective | null |
diagonal_succ_projective :
Projective (diagonal k G (n + 1)) := by
classical
exact Projective.of_iso (diagonalSuccIsoFree k G n).symm inferInstance | instance | 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 | diagonal_succ_projective | null |
leftRegular_projective :
Projective (leftRegular k G) :=
Projective.of_iso (diagonalOneIsoLeftRegular k G) inferInstance | instance | 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 | leftRegular_projective | null |
trivial_projective_of_subsingleton [Subsingleton G] :
Projective (trivial k G k) :=
Projective.of_iso (ofMulActionSubsingletonIsoTrivial _ _ (Fin 1 → G)) diagonal_succ_projective | instance | 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_projective_of_subsingleton | null |
invtSubmodule : Sublattice (Submodule k V) :=
⨅ g, Module.End.invtSubmodule (ρ g) | def | RepresentationTheory | [
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.RepresentationTheory.Basic"
] | Mathlib/RepresentationTheory/Submodule.lean | invtSubmodule | Given a representation `ρ` of a group, `ρ.invtSubmodule` is the sublattice of all
`ρ`-invariant submodules. |
mem_invtSubmodule {p : Submodule k V} :
p ∈ ρ.invtSubmodule ↔ ∀ g, p ∈ Module.End.invtSubmodule (ρ g) := by
rw [invtSubmodule, Sublattice.mem_iInf] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.RepresentationTheory.Basic"
] | Mathlib/RepresentationTheory/Submodule.lean | mem_invtSubmodule | null |
@[simp] protected top_mem : ⊤ ∈ ρ.invtSubmodule := by simp [invtSubmodule]
@[simp] protected lemma bot_mem : ⊥ ∈ ρ.invtSubmodule := by simp [invtSubmodule] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.RepresentationTheory.Basic"
] | Mathlib/RepresentationTheory/Submodule.lean | top_mem | null |
@[simp] protected coe_top : (↑(⊤ : ρ.invtSubmodule) : Submodule k V) = ⊤ := rfl
@[simp] protected lemma coe_bot : (↑(⊥ : ρ.invtSubmodule) : Submodule k V) = ⊥ := rfl | lemma | RepresentationTheory | [
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.RepresentationTheory.Basic"
] | Mathlib/RepresentationTheory/Submodule.lean | coe_top | null |
protected nontrivial_iff : Nontrivial ρ.invtSubmodule ↔ Nontrivial V := by
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· contrapose! h
infer_instance
· refine ⟨⊥, ⊤, ?_⟩
rw [← Subtype.coe_ne_coe, invtSubmodule.coe_top, invtSubmodule.coe_bot]
exact bot_ne_top | lemma | RepresentationTheory | [
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.RepresentationTheory.Basic"
] | Mathlib/RepresentationTheory/Submodule.lean | nontrivial_iff | null |
asAlgebraHom_mem_of_forall_mem (p : Submodule k V) (hp : ∀ g, ∀ v ∈ p, ρ g v ∈ p)
(v : V) (hv : v ∈ p) (x : MonoidAlgebra k G) :
ρ.asAlgebraHom x v ∈ p := by
apply x.induction_on <;> aesop | lemma | RepresentationTheory | [
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.RepresentationTheory.Basic"
] | Mathlib/RepresentationTheory/Submodule.lean | asAlgebraHom_mem_of_forall_mem | null |
noncomputable mapSubmodule :
ρ.invtSubmodule ≃o Submodule (MonoidAlgebra k G) ρ.asModule where
toFun p :=
{ toAddSubmonoid := (p : Submodule k V).toAddSubmonoid.map ρ.asModuleEquiv.symm
smul_mem' := by
simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,
AddSubmonoi... | def | RepresentationTheory | [
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.RepresentationTheory.Basic"
] | Mathlib/RepresentationTheory/Submodule.lean | mapSubmodule | The natural order isomorphism between the two ways to represent invariant submodules. |
forget := LaxMonoidalFunctor.of (forget₂ (FDRep k G) (FGModuleCat k))
@[simp] lemma forget_obj (X : FDRep k G) : (forget k G).obj X = X.V := rfl
@[simp] lemma forget_map (X Y : FDRep k G) (f : X ⟶ Y) : (forget k G).map f = f.hom := rfl | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | forget | The monoidal forgetful functor from `FDRep k G` to `FGModuleCat k`. |
@[simps]
equivApp (g : G) (X : FDRep k G) : X.V ≅ X.V where
hom := ofHom (X.ρ g)
inv := ofHom (X.ρ g⁻¹)
hom_inv_id := by
ext x
simp
inv_hom_id := by
ext x
simp
variable (k G) in | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | equivApp | Definition of `equivHom g : Aut (forget k G)` by its components. |
@[simps]
equivHom : G →* Aut (forget k G) where
toFun g :=
LaxMonoidalFunctor.isoOfComponents (equivApp g) (fun f ↦ (f.comm g).symm) rfl (by intros; rfl)
map_one' := by ext; simp; rfl
map_mul' _ _ := by ext; simp; rfl | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | equivHom | The group homomorphism `G →* Aut (forget k G)` shown to be an isomorphism. |
rightRegular : Representation k G (G → k) where
toFun s :=
{ toFun f t := f (t * s)
map_add' _ _ := rfl
map_smul' _ _ := rfl }
map_one' := by
ext
simp
map_mul' _ _ := by
ext
simp [mul_assoc]
@[simp] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | rightRegular | The representation on `G → k` induced by multiplication on the right in `G`. |
rightRegular_apply (s t : G) (f : G → k) : rightRegular s f t = f (t * s) := rfl | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | rightRegular_apply | null |
leftRegular : Representation k G (G → k) where
toFun s :=
{ toFun f t := f (s⁻¹ * t)
map_add' _ _ := rfl
map_smul' _ _ := rfl }
map_one' := by
ext
simp
map_mul' _ _ := by
ext
simp [mul_assoc]
@[simp] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | leftRegular | The representation on `G → k` induced by multiplication on the left in `G`. |
leftRegular_apply (s t : G) (f : G → k) : leftRegular s f t = f (s⁻¹ * t) := rfl | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | leftRegular_apply | null |
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