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 |
|---|---|---|---|---|---|---|---|---|---|---|
average_map_id (v : V) (hv : v ∈ invariants ρ) : average_map ρ v = v | begin
rw mem_invariants at hv,
simp [average, map_sum, hv, finset.card_univ, nsmul_eq_smul_cast k _ v, smul_smul],
end | theorem | representation.average_map_id | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [
"finset.card_univ",
"nsmul_eq_smul_cast",
"smul_smul"
] | The `average_map` acts as the identity on the subspace of invariants. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_proj_average_map : linear_map.is_proj ρ.invariants ρ.average_map | ⟨ρ.average_map_invariant, ρ.average_map_id⟩ | theorem | representation.is_proj_average_map | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [
"linear_map.is_proj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_invariants_iff_comm {X Y : Rep k G} (f : X.V →ₗ[k] Y.V) (g : G) :
(lin_hom X.ρ Y.ρ) g f = f ↔ f.comp (X.ρ g) = (Y.ρ g).comp f | begin
dsimp,
erw [←ρ_Aut_apply_inv],
rw [←linear_map.comp_assoc, ←Module.comp_def, ←Module.comp_def, iso.inv_comp_eq, ρ_Aut_apply_hom],
exact comm,
end | lemma | representation.lin_hom.mem_invariants_iff_comm | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [
"Rep",
"comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invariants_equiv_Rep_hom (X Y : Rep k G) : (lin_hom X.ρ Y.ρ).invariants ≃ₗ[k] (X ⟶ Y) | { to_fun := λ f, ⟨f.val, λ g, (mem_invariants_iff_comm _ g).1 (f.property g)⟩,
map_add' := λ _ _, rfl,
map_smul' := λ _ _, rfl,
inv_fun := λ f, ⟨f.hom, λ g, (mem_invariants_iff_comm _ g).2 (f.comm g)⟩,
left_inv := λ _, by { ext, refl },
right_inv := λ _, by { ext, refl } } | def | representation.lin_hom.invariants_equiv_Rep_hom | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [
"Rep",
"inv_fun"
] | The invariants of the representation `lin_hom X.ρ Y.ρ` correspond to the the representation
homomorphisms from `X` to `Y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
invariants_equiv_fdRep_hom (X Y : fdRep k G) : (lin_hom X.ρ Y.ρ).invariants ≃ₗ[k] (X ⟶ Y) | begin
rw [←fdRep.forget₂_ρ, ←fdRep.forget₂_ρ],
exact (lin_hom.invariants_equiv_Rep_hom _ _) ≪≫ₗ (fdRep.forget₂_hom_linear_equiv X Y),
end | def | representation.lin_hom.invariants_equiv_fdRep_hom | representation_theory | src/representation_theory/invariants.lean | [
"representation_theory.basic",
"representation_theory.fdRep"
] | [
"fdRep",
"fdRep.forget₂_hom_linear_equiv"
] | The invariants of the representation `lin_hom X.ρ Y.ρ` correspond to the the representation
homomorphisms from `X` to `Y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conjugate (g : G) : W →ₗ[k] V | ((group_smul.linear_map k V g⁻¹).comp π).comp (group_smul.linear_map k W g) | def | linear_map.conjugate | representation_theory | src/representation_theory/maschke.lean | [
"algebra.monoid_algebra.basic",
"algebra.char_p.invertible",
"linear_algebra.basis"
] | [] | We define the conjugate of `π` by `g`, as a `k`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conjugate_i (g : G) (v : V) : (conjugate π g) (i v) = v | begin
dsimp [conjugate],
simp only [←i.map_smul, h, ←mul_smul, single_mul_single, mul_one, mul_left_inv],
change (1 : monoid_algebra k G) • v = v,
simp,
end | lemma | linear_map.conjugate_i | representation_theory | src/representation_theory/maschke.lean | [
"algebra.monoid_algebra.basic",
"algebra.char_p.invertible",
"linear_algebra.basis"
] | [
"monoid_algebra",
"mul_left_inv",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_of_conjugates : W →ₗ[k] V | ∑ g : G, π.conjugate g | def | linear_map.sum_of_conjugates | representation_theory | src/representation_theory/maschke.lean | [
"algebra.monoid_algebra.basic",
"algebra.char_p.invertible",
"linear_algebra.basis"
] | [] | The sum of the conjugates of `π` by each element `g : G`, as a `k`-linear map.
(We postpone dividing by the size of the group as long as possible.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_of_conjugates_equivariant : W →ₗ[monoid_algebra k G] V | monoid_algebra.equivariant_of_linear_of_comm (π.sum_of_conjugates G) (λ g v,
begin
simp only [sum_of_conjugates, linear_map.sum_apply,
-- We have a `module (monoid_algebra k G)` instance but are working with `finsupp`s,
-- so help the elaborator unfold everything correctly.
@finset.smul_sum (monoid_algebr... | def | linear_map.sum_of_conjugates_equivariant | representation_theory | src/representation_theory/maschke.lean | [
"algebra.monoid_algebra.basic",
"algebra.char_p.invertible",
"linear_algebra.basis"
] | [
"finset.smul_sum",
"function.embedding.coe_fn_mk",
"inv_inv",
"inv_mul_cancel_right",
"linear_map.sum_apply",
"monoid_algebra",
"monoid_algebra.equivariant_of_linear_of_comm",
"mul_inv_rev",
"mul_one",
"mul_right_embedding"
] | In fact, the sum over `g : G` of the conjugate of `π` by `g` is a `k[G]`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivariant_projection : W →ₗ[monoid_algebra k G] V | ⅟(fintype.card G : k) • (π.sum_of_conjugates_equivariant G) | def | linear_map.equivariant_projection | representation_theory | src/representation_theory/maschke.lean | [
"algebra.monoid_algebra.basic",
"algebra.char_p.invertible",
"linear_algebra.basis"
] | [
"fintype.card",
"monoid_algebra"
] | We construct our `k[G]`-linear retraction of `i` as
$$ \frac{1}{|G|} \sum_{g \in G} g⁻¹ • π(g • -). $$ | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivariant_projection_condition (v : V) : (π.equivariant_projection G) (i v) = v | begin
rw [equivariant_projection, smul_apply, sum_of_conjugates_equivariant,
equivariant_of_linear_of_comm_apply, sum_of_conjugates],
rw [linear_map.sum_apply],
simp only [conjugate_i π i h],
rw [finset.sum_const, finset.card_univ, nsmul_eq_smul_cast k,
←mul_smul, invertible.inv_of_mul_self, one_smul],
... | lemma | linear_map.equivariant_projection_condition | representation_theory | src/representation_theory/maschke.lean | [
"algebra.monoid_algebra.basic",
"algebra.char_p.invertible",
"linear_algebra.basis"
] | [
"finset.card_univ",
"linear_map.sum_apply",
"nsmul_eq_smul_cast",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_left_inverse_of_injective
(f : V →ₗ[monoid_algebra k G] W) (hf : f.ker = ⊥) :
∃ (g : W →ₗ[monoid_algebra k G] V), g.comp f = linear_map.id | begin
obtain ⟨φ, hφ⟩ := (f.restrict_scalars k).exists_left_inverse_of_injective
(by simp only [hf, submodule.restrict_scalars_bot, linear_map.ker_restrict_scalars]),
refine ⟨φ.equivariant_projection G, _⟩,
apply linear_map.ext,
intro v,
simp only [linear_map.id_coe, id.def, linear_map.comp_apply],
apply... | lemma | monoid_algebra.exists_left_inverse_of_injective | representation_theory | src/representation_theory/maschke.lean | [
"algebra.monoid_algebra.basic",
"algebra.char_p.invertible",
"linear_algebra.basis"
] | [
"linear_map.comp_apply",
"linear_map.equivariant_projection_condition",
"linear_map.ext",
"linear_map.id",
"linear_map.id_coe",
"linear_map.ker_restrict_scalars",
"monoid_algebra",
"submodule.restrict_scalars_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_is_compl
(p : submodule (monoid_algebra k G) V) :
∃ q : submodule (monoid_algebra k G) V, is_compl p q | let ⟨f, hf⟩ := monoid_algebra.exists_left_inverse_of_injective p.subtype p.ker_subtype in
⟨f.ker, linear_map.is_compl_of_proj $ linear_map.ext_iff.1 hf⟩ | lemma | monoid_algebra.submodule.exists_is_compl | representation_theory | src/representation_theory/maschke.lean | [
"algebra.monoid_algebra.basic",
"algebra.char_p.invertible",
"linear_algebra.basis"
] | [
"is_compl",
"linear_map.is_compl_of_proj",
"monoid_algebra",
"monoid_algebra.exists_left_inverse_of_injective",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complemented_lattice : complemented_lattice (submodule (monoid_algebra k G) V) | ⟨exists_is_compl⟩ | instance | monoid_algebra.submodule.complemented_lattice | representation_theory | src/representation_theory/maschke.lean | [
"algebra.monoid_algebra.basic",
"algebra.char_p.invertible",
"linear_algebra.basis"
] | [
"complemented_lattice",
"monoid_algebra",
"submodule"
] | This also implies an instance `is_semisimple_module (monoid_algebra k G) V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Rep (k G : Type u) [ring k] [monoid G] | Action (Module.{u} k) (Mon.of G) | abbreviation | Rep | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Action",
"Mon.of",
"monoid",
"ring"
] | The category of `k`-linear representations of a monoid `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ρ (V : Rep k G) : representation k G V | V.ρ | def | Rep.ρ | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"representation"
] | Specialize the existing `Action.ρ`, changing the type to `representation k G V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of {V : Type u} [add_comm_group V] [module k V] (ρ : G →* (V →ₗ[k] V)) : Rep k G | ⟨Module.of k V, ρ⟩ | def | Rep.of | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"add_comm_group",
"module"
] | Lift an unbundled representation to `Rep`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of {V : Type u} [add_comm_group V] [module k V] (ρ : G →* (V →ₗ[k] V)) :
(of ρ : Type u) = V | rfl | lemma | Rep.coe_of | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"add_comm_group",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_ρ {V : Type u} [add_comm_group V] [module k V] (ρ : G →* (V →ₗ[k] V)) :
(of ρ).ρ = ρ | rfl | lemma | Rep.of_ρ | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"add_comm_group",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Action_ρ_eq_ρ {A : Rep k G} : Action.ρ A = A.ρ | rfl | lemma | Rep.Action_ρ_eq_ρ | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_ρ_apply {V : Type u} [add_comm_group V] [module k V]
(ρ : representation k G V) (g : Mon.of G) :
(Rep.of ρ).ρ g = ρ (g : G) | rfl | lemma | Rep.of_ρ_apply | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Mon.of",
"Rep.of",
"add_comm_group",
"module",
"representation"
] | Allows us to apply lemmas about the underlying `ρ`, which would take an element `g : G` rather
than `g : Mon.of G` as an argument. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ρ_inv_self_apply {G : Type u} [group G] (A : Rep k G) (g : G) (x : A) :
A.ρ g⁻¹ (A.ρ g x) = x | show (A.ρ g⁻¹ * A.ρ g) x = x, by rw [←map_mul, inv_mul_self, map_one, linear_map.one_apply] | lemma | Rep.ρ_inv_self_apply | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"group",
"inv_mul_self",
"linear_map.one_apply",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ρ_self_inv_apply {G : Type u} [group G] {A : Rep k G} (g : G) (x : A) :
A.ρ g (A.ρ g⁻¹ x) = x | show (A.ρ g * A.ρ g⁻¹) x = x, by rw [←map_mul, mul_inv_self, map_one, linear_map.one_apply] | lemma | Rep.ρ_self_inv_apply | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"group",
"linear_map.one_apply",
"map_one",
"mul_inv_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_comm_apply {A B : Rep k G} (f : A ⟶ B) (g : G) (x : A) :
f.hom (A.ρ g x) = B.ρ g (f.hom x) | linear_map.ext_iff.1 (f.comm g) x | lemma | Rep.hom_comm_apply | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivial (V : Type u) [add_comm_group V] [module k V] : Rep k G | Rep.of (@representation.trivial k G V _ _ _ _) | def | Rep.trivial | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"Rep.of",
"add_comm_group",
"module",
"representation.trivial"
] | The trivial `k`-linear `G`-representation on a `k`-module `V.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivial_def {V : Type u} [add_comm_group V] [module k V] (g : G) (v : V) :
(trivial k G V).ρ g v = v | rfl | lemma | Rep.trivial_def | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"add_comm_group",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoidal_category.braiding_hom_apply {A B : Rep k G} (x : A) (y : B) :
Action.hom.hom (β_ A B).hom (tensor_product.tmul k x y) = tensor_product.tmul k y x | rfl | lemma | Rep.monoidal_category.braiding_hom_apply | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"tensor_product.tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoidal_category.braiding_inv_apply {A B : Rep k G} (x : A) (y : B) :
Action.hom.hom (β_ A B).inv (tensor_product.tmul k y x) = tensor_product.tmul k x y | rfl | lemma | Rep.monoidal_category.braiding_inv_apply | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"tensor_product.tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linearization :
monoidal_functor (Action (Type u) (Mon.of G)) (Rep k G) | (Module.monoidal_free k).map_Action (Mon.of G) | def | Rep.linearization | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Action",
"Module.monoidal_free",
"Mon.of",
"Rep"
] | The monoidal functor sending a type `H` with a `G`-action to the induced `k`-linear
`G`-representation on `k[H].` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linearization_obj_ρ (X : Action (Type u) (Mon.of G)) (g : G) (x : X.V →₀ k) :
((linearization k G).obj X).ρ g x = finsupp.lmap_domain k k (X.ρ g) x | rfl | lemma | Rep.linearization_obj_ρ | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Action",
"Mon.of",
"finsupp.lmap_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linearization_of (X : Action (Type u) (Mon.of 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 rw [linearization_obj_ρ, finsupp.lmap_domain_apply, finsupp.map_domain_single] | lemma | Rep.linearization_of | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Action",
"Mon.of",
"finsupp.lmap_domain_apply",
"finsupp.map_domain_single",
"finsupp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linearization_map_hom :
((linearization k G).map f).hom = finsupp.lmap_domain k k f.hom | rfl | lemma | Rep.linearization_map_hom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"finsupp.lmap_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | by rw [linearization_map_hom, finsupp.lmap_domain_apply, finsupp.map_domain_single] | lemma | Rep.linearization_map_hom_single | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"finsupp.lmap_domain_apply",
"finsupp.map_domain_single",
"finsupp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linearization_μ_hom (X Y : Action (Type u) (Mon.of G)) :
((linearization k G).μ X Y).hom = (finsupp_tensor_finsupp' k X.V Y.V).to_linear_map | rfl | lemma | Rep.linearization_μ_hom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Action",
"Mon.of",
"finsupp_tensor_finsupp'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linearization_μ_inv_hom (X Y : Action (Type u) (Mon.of G)) :
(inv ((linearization k G).μ X Y)).hom = (finsupp_tensor_finsupp' k X.V Y.V).symm.to_linear_map | begin
simp_rw [←Action.forget_map, functor.map_inv, Action.forget_map, linearization_μ_hom],
apply is_iso.inv_eq_of_hom_inv_id _,
exact linear_map.ext (λ x, linear_equiv.symm_apply_apply _ _),
end | lemma | Rep.linearization_μ_inv_hom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Action",
"Mon.of",
"finsupp_tensor_finsupp'",
"linear_equiv.symm_apply_apply",
"linear_map.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linearization_ε_hom :
(linearization k G).ε.hom = finsupp.lsingle punit.star | rfl | lemma | Rep.linearization_ε_hom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"finsupp.lsingle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linearization_ε_inv_hom_apply (r : k) :
(inv (linearization k G).ε).hom (finsupp.single punit.star r) = r | begin
simp_rw [←Action.forget_map, functor.map_inv, Action.forget_map],
rw [←finsupp.lsingle_apply punit.star r],
apply is_iso.hom_inv_id_apply _ _,
end | lemma | Rep.linearization_ε_inv_hom_apply | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"finsupp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linearization_trivial_iso (X : Type u) :
(linearization k G).obj (Action.mk X 1) ≅ trivial k G (X →₀ k) | Action.mk_iso (iso.refl _) $ λ g, by { ext1, ext1, exact linearization_of _ _ _ } | def | Rep.linearization_trivial_iso | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Action.mk_iso"
] | The linearization of a type `X` on which `G` acts trivially is the trivial `G`-representation
on `k[X]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_mul_action (H : Type u) [mul_action G H] : Rep k G | of $ representation.of_mul_action k G H | abbreviation | Rep.of_mul_action | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"mul_action",
"representation.of_mul_action"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_regular : Rep k G | of_mul_action k G G | def | Rep.left_regular | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep"
] | The `k`-linear `G`-representation on `k[G]`, induced by left multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal (n : ℕ) : Rep k G | of_mul_action k G (fin n → G) | def | Rep.diagonal | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep"
] | The `k`-linear `G`-representation on `k[Gⁿ]`, induced by left multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linearization_of_mul_action_iso (H : Type u) [mul_action G H] :
(linearization k G).obj (Action.of_mul_action G H)
≅ of_mul_action k G H | iso.refl _ | def | Rep.linearization_of_mul_action_iso | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Action.of_mul_action",
"mul_action"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_regular_hom (A : Rep k G) (x : A) :
Rep.of_mul_action k G G ⟶ A | { hom := finsupp.lift _ _ _ (λ g, A.ρ g x),
comm' := λ g,
begin
refine finsupp.lhom_ext' (λ y, linear_map.ext_ring _),
simpa only [linear_map.comp_apply, Module.comp_def, finsupp.lsingle_apply,
finsupp.lift_apply, Action_ρ_eq_ρ, of_ρ_apply, representation.of_mul_action_single,
finsupp.sum_single... | def | Rep.left_regular_hom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Module.comp_def",
"Rep",
"Rep.of_mul_action",
"finsupp.lhom_ext'",
"finsupp.lift",
"finsupp.lift_apply",
"finsupp.lsingle_apply",
"linear_map.comp_apply",
"linear_map.ext_ring",
"one_smul",
"representation.of_mul_action_single",
"smul_eq_mul",
"zero_smul"
] | Given an element `x : A`, there is a natural morphism of representations `k[G] ⟶ A` sending
`g ↦ A.ρ(g)(x).` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_regular_hom_apply {A : Rep k G} (x : A) :
(left_regular_hom A x).hom (finsupp.single 1 1) = x | begin
simpa only [left_regular_hom_hom, finsupp.lift_apply, finsupp.sum_single_index, one_smul,
A.ρ.map_one, zero_smul],
end | lemma | Rep.left_regular_hom_apply | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"finsupp.lift_apply",
"finsupp.single",
"one_smul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_regular_hom_equiv (A : Rep k G) :
(Rep.of_mul_action k G G ⟶ A) ≃ₗ[k] A | { to_fun := λ f, f.hom (finsupp.single 1 1),
map_add' := λ x y, rfl,
map_smul' := λ r x, rfl,
inv_fun := λ x, left_regular_hom A x,
left_inv := λ f,
begin
refine Action.hom.ext _ _ (finsupp.lhom_ext' (λ (x : G), linear_map.ext_ring _)),
have : f.hom (((of_mul_action k G G).ρ) x (finsupp.single (1 : G)... | def | Rep.left_regular_hom_equiv | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"Rep.of_mul_action",
"finsupp.lhom_ext'",
"finsupp.lift_apply",
"finsupp.lsingle_apply",
"finsupp.single",
"inv_fun",
"linear_map.comp_apply",
"linear_map.ext_ring",
"mul_one",
"one_smul",
"representation.of_mul_action_single",
"smul_eq_mul",
"zero_smul"
] | Given a `k`-linear `G`-representation `A`, there is a `k`-linear isomorphism between
representation morphisms `Hom(k[G], A)` and `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_regular_hom_equiv_symm_single {A : Rep k G} (x : A) (g : G) :
((left_regular_hom_equiv A).symm x).hom (finsupp.single g 1) = A.ρ g x | begin
simp only [left_regular_hom_equiv_symm_apply, left_regular_hom_hom,
finsupp.lift_apply, finsupp.sum_single_index, zero_smul, one_smul],
end | lemma | Rep.left_regular_hom_equiv_symm_single | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"finsupp.lift_apply",
"finsupp.single",
"one_smul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ihom (A : Rep k G) : Rep k G ⥤ Rep k G | { obj := λ B, Rep.of (representation.lin_hom A.ρ B.ρ),
map := λ X Y f,
{ hom := Module.of_hom (linear_map.llcomp k _ _ _ f.hom),
comm' := λ g, linear_map.ext (λ x, linear_map.ext (λ y,
show f.hom (X.ρ g _) = _, by simpa only [hom_comm_apply])) },
map_id' := λ B, by ext; refl,
map_comp' := λ B C ... | def | Rep.ihom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Module.of_hom",
"Rep",
"Rep.of",
"linear_map.ext",
"linear_map.llcomp",
"representation.lin_hom"
] | 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)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ihom_obj_ρ_apply {A B : Rep k G} (g : G) (x : A →ₗ[k] B) :
((Rep.ihom A).obj B).ρ g x = (B.ρ g) ∘ₗ x ∘ₗ (A.ρ g⁻¹) | rfl | lemma | Rep.ihom_obj_ρ_apply | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"Rep.ihom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv (A B C : Rep k G) : (A ⊗ B ⟶ C) ≃ (B ⟶ (Rep.ihom A).obj C) | { to_fun := λ f,
{ hom := (tensor_product.curry f.hom).flip,
comm' := λ g,
begin
refine linear_map.ext (λ x, linear_map.ext (λ y, _)),
change f.hom (_ ⊗ₜ[k] _) = C.ρ g (f.hom (_ ⊗ₜ[k] _)),
rw [←hom_comm_apply],
change _ = f.hom ((A.ρ g * A.ρ g⁻¹) y ⊗ₜ[k] _),
simpa onl... | def | Rep.hom_equiv | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Module.comp_def",
"Module.monoidal_category.hom_apply",
"Rep",
"Rep.ihom",
"Rep.ihom_obj_ρ_apply",
"inv_fun",
"linear_map.comp_apply",
"linear_map.ext",
"linear_map.flip_apply",
"map_one",
"mul_inv_self",
"tensor_product.curry",
"tensor_product.ext'",
"tensor_product.map_tmul",
"tensor_... | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ihom_obj_ρ_def (A B : Rep k G) : ((ihom A).obj B).ρ = ((Rep.ihom A).obj B).ρ | rfl | lemma | Rep.ihom_obj_ρ_def | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"Rep.ihom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv_def (A B C : Rep k G) :
(ihom.adjunction A).hom_equiv B C = Rep.hom_equiv A B C | rfl | lemma | Rep.hom_equiv_def | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"Rep.hom_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ihom_ev_app_hom (A B : Rep k G) :
Action.hom.hom ((ihom.ev A).app B)
= tensor_product.uncurry _ _ _ _ linear_map.id.flip | by ext; refl | lemma | Rep.ihom_ev_app_hom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"tensor_product.uncurry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ihom_coev_app_hom (A B : Rep k G) :
Action.hom.hom ((ihom.coev A).app B) = (tensor_product.mk _ _ _).flip | by ext; refl | lemma | Rep.ihom_coev_app_hom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"tensor_product.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoidal_closed.linear_hom_equiv :
(A ⊗ B ⟶ C) ≃ₗ[k] (B ⟶ (A ⟶[Rep k G] C)) | { map_add' := λ f g, rfl,
map_smul' := λ r f, rfl, ..(ihom.adjunction A).hom_equiv _ _ } | def | Rep.monoidal_closed.linear_hom_equiv | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep"
] | There is a `k`-linear isomorphism between the sets of representation morphisms`Hom(A ⊗ B, C)`
and `Hom(B, Homₖ(A, C))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoidal_closed.linear_hom_equiv_comm :
(A ⊗ B ⟶ C) ≃ₗ[k] (A ⟶ (B ⟶[Rep k G] C)) | (linear.hom_congr k (β_ A B) (iso.refl _)) ≪≫ₗ
monoidal_closed.linear_hom_equiv _ _ _ | def | Rep.monoidal_closed.linear_hom_equiv_comm | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep"
] | There is a `k`-linear isomorphism between the sets of representation morphisms`Hom(A ⊗ B, C)`
and `Hom(A, Homₖ(B, C))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoidal_closed.linear_hom_equiv_hom (f : A ⊗ B ⟶ C) :
(monoidal_closed.linear_hom_equiv A B C f).hom =
(tensor_product.curry f.hom).flip | rfl | lemma | Rep.monoidal_closed.linear_hom_equiv_hom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"tensor_product.curry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoidal_closed.linear_hom_equiv_comm_hom (f : A ⊗ B ⟶ C) :
(monoidal_closed.linear_hom_equiv_comm A B C f).hom =
tensor_product.curry f.hom | rfl | lemma | Rep.monoidal_closed.linear_hom_equiv_comm_hom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"tensor_product.curry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoidal_closed.linear_hom_equiv_symm_hom (f : B ⟶ (A ⟶[Rep k G] C)) :
((monoidal_closed.linear_hom_equiv A B C).symm f).hom =
tensor_product.uncurry k A B C f.hom.flip | rfl | lemma | Rep.monoidal_closed.linear_hom_equiv_symm_hom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"tensor_product.uncurry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoidal_closed.linear_hom_equiv_comm_symm_hom (f : A ⟶ (B ⟶[Rep k G] C)) :
((monoidal_closed.linear_hom_equiv_comm A B C).symm f).hom =
tensor_product.uncurry k A B C f.hom | by ext; refl | lemma | Rep.monoidal_closed.linear_hom_equiv_comm_symm_hom | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"tensor_product.uncurry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Rep_of_tprod_iso : Rep.of (ρ.tprod τ) ≅ Rep.of ρ ⊗ Rep.of τ | iso.refl _ | def | representation.Rep_of_tprod_iso | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep.of"
] | Tautological isomorphism to help Lean in typechecking. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Rep_of_tprod_iso_apply (x : tensor_product k V W) :
(Rep_of_tprod_iso ρ τ).hom.hom x = x | rfl | lemma | representation.Rep_of_tprod_iso_apply | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"tensor_product"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Rep_of_tprod_iso_inv_apply (x : tensor_product k V W) :
(Rep_of_tprod_iso ρ τ).inv.hom x = x | rfl | lemma | representation.Rep_of_tprod_iso_inv_apply | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"tensor_product"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Module_monoid_algebra_map_aux {k G : Type*} [comm_ring k] [monoid G]
(V W : Type*) [add_comm_group V] [add_comm_group 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 : monoid_algebra k G) (x : V) :
f ((((monoid_algebra.l... | begin
apply monoid_algebra.induction_on r,
{ intro g,
simp only [one_smul, monoid_algebra.lift_single, monoid_algebra.of_apply],
exact linear_map.congr_fun (w g) x, },
{ intros g h gw hw, simp only [map_add, add_left_inj, linear_map.add_apply, hw, gw], },
{ intros r g w,
simp only [alg_hom.map_smul,... | lemma | Rep.to_Module_monoid_algebra_map_aux | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"add_comm_group",
"alg_hom.map_smul",
"comm_ring",
"linear_map.add_apply",
"linear_map.congr_fun",
"linear_map.map_smulₛₗ",
"linear_map.smul_apply",
"module",
"monoid",
"monoid_algebra",
"monoid_algebra.induction_on",
"monoid_algebra.lift",
"monoid_algebra.lift_single",
"one_smul",
"ring... | Auxilliary lemma for `to_Module_monoid_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Module_monoid_algebra_map {V W : Rep k G} (f : V ⟶ W) :
Module.of (monoid_algebra k G) V.ρ.as_module ⟶ Module.of (monoid_algebra k G) W.ρ.as_module | { map_smul' := λ r x, to_Module_monoid_algebra_map_aux V.V W.V V.ρ W.ρ f.hom f.comm r x,
..f.hom, } | def | Rep.to_Module_monoid_algebra_map | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Module.of",
"Rep",
"monoid_algebra"
] | Auxilliary definition for `to_Module_monoid_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Module_monoid_algebra : Rep k G ⥤ Module.{u} (monoid_algebra k G) | { obj := λ V, Module.of _ V.ρ.as_module ,
map := λ V W f, to_Module_monoid_algebra_map f, } | def | Rep.to_Module_monoid_algebra | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Module.of",
"Rep",
"monoid_algebra"
] | Functorially convert a representation of `G` into a module over `monoid_algebra k G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_Module_monoid_algebra : Module.{u} (monoid_algebra k G) ⥤ Rep k G | { obj := λ M, Rep.of (representation.of_module k G M),
map := λ M N f,
{ hom :=
{ map_smul' := λ r x, f.map_smul (algebra_map k _ r) x,
..f },
comm' := λ g, by { ext, apply f.map_smul, }, }, }. | def | Rep.of_Module_monoid_algebra | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"Rep.of",
"algebra_map",
"monoid_algebra",
"representation.of_module"
] | Functorially convert a module over `monoid_algebra k G` into a representation of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_Module_monoid_algebra_obj_coe (M : Module.{u} (monoid_algebra k G)) :
(of_Module_monoid_algebra.obj M : Type u) = restrict_scalars k (monoid_algebra k G) M | rfl | lemma | Rep.of_Module_monoid_algebra_obj_coe | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"monoid_algebra",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_Module_monoid_algebra_obj_ρ (M : Module.{u} (monoid_algebra k G)) :
(of_Module_monoid_algebra.obj M).ρ = representation.of_module k G M | rfl | lemma | Rep.of_Module_monoid_algebra_obj_ρ | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"monoid_algebra",
"representation.of_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
counit_iso_add_equiv {M : Module.{u} (monoid_algebra k G)} :
((of_Module_monoid_algebra ⋙ to_Module_monoid_algebra).obj M) ≃+ M | begin
dsimp [of_Module_monoid_algebra, to_Module_monoid_algebra],
refine (representation.of_module k G ↥M).as_module_equiv.trans (restrict_scalars.add_equiv _ _ _),
end | def | Rep.counit_iso_add_equiv | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"monoid_algebra",
"representation.of_module",
"restrict_scalars.add_equiv"
] | Auxilliary definition for `equivalence_Module_monoid_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_iso_add_equiv {V : Rep k G} :
V ≃+ ((to_Module_monoid_algebra ⋙ of_Module_monoid_algebra).obj V) | begin
dsimp [of_Module_monoid_algebra, to_Module_monoid_algebra],
refine V.ρ.as_module_equiv.symm.trans _,
exact (restrict_scalars.add_equiv _ _ _).symm,
end | def | Rep.unit_iso_add_equiv | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"restrict_scalars.add_equiv"
] | Auxilliary definition for `equivalence_Module_monoid_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit_iso (M : Module.{u} (monoid_algebra k G)) :
(of_Module_monoid_algebra ⋙ to_Module_monoid_algebra).obj M ≅ M | linear_equiv.to_Module_iso'
{ map_smul' := λ r x, begin
dsimp [counit_iso_add_equiv],
simp,
end,
..counit_iso_add_equiv, } | def | Rep.counit_iso | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"linear_equiv.to_Module_iso'",
"monoid_algebra"
] | Auxilliary definition for `equivalence_Module_monoid_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_iso_comm (V : Rep k G) (g : G) (x : V) :
unit_iso_add_equiv (((V.ρ) g).to_fun x) =
(((of_Module_monoid_algebra.obj (to_Module_monoid_algebra.obj V)).ρ) g).to_fun
(unit_iso_add_equiv x) | begin
dsimp [unit_iso_add_equiv, of_Module_monoid_algebra, to_Module_monoid_algebra],
simp only [add_equiv.apply_eq_iff_eq, add_equiv.apply_symm_apply,
representation.as_module_equiv_symm_map_rho, representation.of_module_as_module_act],
end | lemma | Rep.unit_iso_comm | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"representation.as_module_equiv_symm_map_rho",
"representation.of_module_as_module_act"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_iso (V : Rep k G) :
V ≅ ((to_Module_monoid_algebra ⋙ of_Module_monoid_algebra).obj V) | Action.mk_iso (linear_equiv.to_Module_iso'
{ map_smul' := λ r x, begin
dsimp [unit_iso_add_equiv],
simp only [representation.as_module_equiv_symm_map_smul,
restrict_scalars.add_equiv_symm_map_algebra_map_smul],
end,
..unit_iso_add_equiv, })
(λ g, by { ext, apply unit_iso_comm, }) | def | Rep.unit_iso | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Action.mk_iso",
"Rep",
"linear_equiv.to_Module_iso'",
"representation.as_module_equiv_symm_map_smul",
"restrict_scalars.add_equiv_symm_map_algebra_map_smul"
] | Auxilliary definition for `equivalence_Module_monoid_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_Module_monoid_algebra : Rep k G ≌ Module.{u} (monoid_algebra k G) | { functor := to_Module_monoid_algebra,
inverse := of_Module_monoid_algebra,
unit_iso := nat_iso.of_components (λ V, unit_iso V) (by tidy),
counit_iso := nat_iso.of_components (λ M, counit_iso M) (by tidy), } | def | Rep.equivalence_Module_monoid_algebra | representation_theory | src/representation_theory/Rep.lean | [
"representation_theory.basic",
"representation_theory.Action",
"algebra.category.Module.abelian",
"algebra.category.Module.colimits",
"algebra.category.Module.monoidal.closed",
"algebra.category.Module.adjunctions",
"category_theory.closed.functor_category"
] | [
"Rep",
"monoid_algebra"
] | The categorical equivalence `Rep k G ≌ Module (monoid_algebra k G)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_yoneda_obj_resolution (A : Rep k G) : cochain_complex (Module.{u} k) ℕ | homological_complex.unop
((((linear_yoneda k (Rep k G)).obj A).right_op.map_homological_complex _).obj (resolution k G)) | abbreviation | group_cohomology.linear_yoneda_obj_resolution | representation_theory.group_cohomology | src/representation_theory/group_cohomology/basic.lean | [
"algebra.homology.opposite",
"representation_theory.group_cohomology.resolution"
] | [
"Rep",
"cochain_complex",
"homological_complex.unop"
] | The complex `Hom(P, A)`, where `P` is the standard resolution of `k` as a trivial `k`-linear
`G`-representation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_yoneda_obj_resolution_d_apply {A : Rep k G} (i j : ℕ) (x : (resolution k G).X i ⟶ A) :
(linear_yoneda_obj_resolution A).d i j x = (resolution k G).d j i ≫ x | rfl | lemma | group_cohomology.linear_yoneda_obj_resolution_d_apply | representation_theory.group_cohomology | src/representation_theory/group_cohomology/basic.lean | [
"algebra.homology.opposite",
"representation_theory.group_cohomology.resolution"
] | [
"Rep"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d [monoid G] (n : ℕ) (A : Rep k G) :
((fin n → G) → A) →ₗ[k] (fin (n + 1) → G) → A | { to_fun := λ f g, A.ρ (g 0) (f (λ i, g i.succ))
+ finset.univ.sum (λ j : fin (n + 1), (-1 : k) ^ ((j : ℕ) + 1) • f (fin.contract_nth j (*) g)),
map_add' := λ f g,
begin
ext x,
simp only [pi.add_apply, map_add, smul_add, finset.sum_add_distrib, add_add_add_comm],
end,
map_smul' := λ r f,
begin
... | def | inhomogeneous_cochains.d | representation_theory.group_cohomology | src/representation_theory/group_cohomology/basic.lean | [
"algebra.homology.opposite",
"representation_theory.group_cohomology.resolution"
] | [
"Rep",
"fin.contract_nth",
"finset.smul_sum",
"monoid",
"mul_comm",
"pi.smul_apply",
"ring_hom.id_apply",
"smul_add",
"smul_eq_mul"
] | The differential in the complex of inhomogeneous cochains used to
calculate group cohomology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_eq :
d n A = ((diagonal_hom_equiv n A).to_Module_iso.inv
≫ (linear_yoneda_obj_resolution A).d n (n + 1)
≫ (diagonal_hom_equiv (n + 1) A).to_Module_iso.hom) | begin
ext f g,
simp only [Module.coe_comp, linear_equiv.coe_coe, function.comp_app,
linear_equiv.to_Module_iso_inv, linear_yoneda_obj_resolution_d_apply,
linear_equiv.to_Module_iso_hom, diagonal_hom_equiv_apply, Action.comp_hom,
resolution.d_eq k G n, resolution.d_of (fin.partial_prod g), linear_map.map... | lemma | inhomogeneous_cochains.d_eq | representation_theory.group_cohomology | src/representation_theory/group_cohomology/basic.lean | [
"algebra.homology.opposite",
"representation_theory.group_cohomology.resolution"
] | [
"Action.comp_hom",
"Module.coe_comp",
"fin.cast_succ",
"fin.cast_succ_fin_succ",
"fin.cast_succ_zero",
"fin.coe_succ",
"fin.coe_zero",
"fin.partial_prod",
"fin.partial_prod_right_inv",
"fin.partial_prod_succ",
"fin.partial_prod_zero",
"fin.succ_above_zero",
"inv_mul_cancel_left",
"linear_e... | The theorem that our isomorphism `Fun(Gⁿ, A) ≅ Hom(k[Gⁿ⁺¹], A)` (where the righthand side is
morphisms in `Rep k G`) commutes with the differentials in the complex of inhomogeneous cochains
and the homogeneous `linear_yoneda_obj_resolution`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhomogeneous_cochains : cochain_complex (Module k) ℕ | cochain_complex.of (λ n, Module.of k ((fin n → G) → A))
(λ n, inhomogeneous_cochains.d n A) (λ n,
begin
ext x y,
have := linear_map.ext_iff.1 ((linear_yoneda_obj_resolution A).d_comp_d n (n + 1) (n + 2)),
simp only [Module.coe_comp, function.comp_app] at this,
simp only [Module.coe_comp, function.comp_app, d_eq... | abbreviation | group_cohomology.inhomogeneous_cochains | representation_theory.group_cohomology | src/representation_theory/group_cohomology/basic.lean | [
"algebra.homology.opposite",
"representation_theory.group_cohomology.resolution"
] | [
"Module",
"Module.coe_comp",
"Module.of",
"cochain_complex",
"cochain_complex.of",
"inhomogeneous_cochains.d",
"linear_equiv.coe_coe",
"linear_equiv.symm_apply_apply",
"linear_map.zero_apply"
] | Given a `k`-linear `G`-representation `A`, this is the complex of inhomogeneous cochains
$$0 \to \mathrm{Fun}(G^0, A) \to \mathrm{Fun}(G^1, A) \to \mathrm{Fun}(G^2, A) \to \dots$$
which calculates the group cohomology of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhomogeneous_cochains_iso :
inhomogeneous_cochains A ≅ linear_yoneda_obj_resolution A | homological_complex.hom.iso_of_components
(λ i, (Rep.diagonal_hom_equiv i A).to_Module_iso.symm) $
begin
rintros i j (h : i + 1 = j),
subst h,
simp only [cochain_complex.of_d, d_eq, category.assoc, iso.symm_hom,
iso.hom_inv_id, category.comp_id],
end | def | group_cohomology.inhomogeneous_cochains_iso | representation_theory.group_cohomology | src/representation_theory/group_cohomology/basic.lean | [
"algebra.homology.opposite",
"representation_theory.group_cohomology.resolution"
] | [
"Rep.diagonal_hom_equiv",
"cochain_complex.of_d",
"homological_complex.hom.iso_of_components"
] | Given a `k`-linear `G`-representation `A`, the complex of inhomogeneous cochains is isomorphic
to `Hom(P, A)`, where `P` is the standard resolution of `k` as a trivial `G`-representation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_cohomology [group G] (A : Rep k G) (n : ℕ) : Module k | (inhomogeneous_cochains A).homology n | def | group_cohomology | representation_theory.group_cohomology | src/representation_theory/group_cohomology/basic.lean | [
"algebra.homology.opposite",
"representation_theory.group_cohomology.resolution"
] | [
"Module",
"Rep",
"group",
"homology"
] | The group cohomology of a `k`-linear `G`-representation `A`, as the cohomology of its complex
of inhomogeneous cochains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_cohomology_iso_Ext [group G] (A : Rep k G) (n : ℕ) :
group_cohomology A n ≅ ((Ext k (Rep k G) n).obj
(opposite.op $ Rep.trivial k G k)).obj A | (homology_obj_iso_of_homotopy_equiv (homotopy_equiv.of_iso (inhomogeneous_cochains_iso _)) _)
≪≫ (homological_complex.homology_unop _ _) ≪≫ (Ext_iso k G A n).symm | def | group_cohomology_iso_Ext | representation_theory.group_cohomology | src/representation_theory/group_cohomology/basic.lean | [
"algebra.homology.opposite",
"representation_theory.group_cohomology.resolution"
] | [
"Ext",
"Rep",
"Rep.trivial",
"group",
"group_cohomology",
"homological_complex.homology_unop",
"homology_obj_iso_of_homotopy_equiv",
"homotopy_equiv.of_iso",
"opposite.op"
] | The `n`th group cohomology of a `k`-linear `G`-representation `A` is isomorphic to
`Extⁿ(k, A)` (taken in `Rep k G`), where `k` is a trivial `k`-linear `G`-representation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Action_diagonal_succ (G : Type u) [group G] : Π (n : ℕ),
diagonal G (n + 1) ≅ left_regular G ⊗ Action.mk (fin n → G) 1 | | 0 := diagonal_one_iso_left_regular G ≪≫ (ρ_ _).symm ≪≫ tensor_iso (iso.refl _)
(tensor_unit_iso (equiv.equiv_of_unique punit _).to_iso)
| (n+1) := diagonal_succ _ _ ≪≫ tensor_iso (iso.refl _) (Action_diagonal_succ n)
≪≫ left_regular_tensor_iso _ _ ≪≫ tensor_iso (iso.refl _) (mk_iso
(equiv.pi_fin_succ_above_equi... | def | group_cohomology.resolution.Action_diagonal_succ | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"equiv.equiv_of_unique",
"equiv.pi_fin_succ_above_equiv",
"group"
] | An isomorphism of `G`-sets `Gⁿ⁺¹ ≅ G × Gⁿ`, where `G` acts by left multiplication on `Gⁿ⁺¹` and
`G` but trivially on `Gⁿ`. The map sends `(g₀, ..., gₙ) ↦ (g₀, (g₀⁻¹g₁, g₁⁻¹g₂, ..., gₙ₋₁⁻¹gₙ))`,
and the inverse is `(g₀, (g₁, ..., gₙ)) ↦ (g₀, g₀g₁, g₀g₁g₂, ..., g₀g₁...gₙ).` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Action_diagonal_succ_hom_apply {G : Type u} [group G] {n : ℕ} (f : fin (n + 1) → G) :
(Action_diagonal_succ G n).hom.hom f = (f 0, λ i, (f i.cast_succ)⁻¹ * f i.succ) | begin
induction n with n hn,
{ exact prod.ext rfl (funext $ λ x, fin.elim0 x) },
{ ext,
{ refl },
{ dunfold Action_diagonal_succ,
simp only [iso.trans_hom, comp_hom, types_comp_apply, diagonal_succ_hom_hom,
left_regular_tensor_iso_hom_hom, tensor_iso_hom, mk_iso_hom_hom, equiv.to_iso_hom,
... | lemma | group_cohomology.resolution.Action_diagonal_succ_hom_apply | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"equiv.to_iso_hom",
"fin.cases",
"fin.cast_succ_fin_succ",
"fin.cons_succ",
"fin.cons_zero",
"fin.insert_nth_zero'",
"group",
"inv_inj",
"monoid_hom.one_apply",
"mul_left_inj",
"prod.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Action_diagonal_succ_inv_apply {G : Type u} [group G] {n : ℕ} (g : G) (f : fin n → G) :
(Action_diagonal_succ G n).inv.hom (g, f) = (g • fin.partial_prod f : fin (n + 1) → G) | begin
revert g,
induction n with n hn,
{ intros g,
ext,
simpa only [subsingleton.elim x 0, pi.smul_apply, fin.partial_prod_zero,
smul_eq_mul, mul_one] },
{ intro g,
ext,
dunfold Action_diagonal_succ,
simp only [iso.trans_inv, comp_hom, hn, diagonal_succ_inv_hom, types_comp_apply,
... | lemma | group_cohomology.resolution.Action_diagonal_succ_inv_apply | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"fin.cases",
"fin.cons_succ",
"fin.cons_zero",
"fin.partial_prod",
"fin.partial_prod_succ'",
"fin.partial_prod_zero",
"group",
"mul_assoc",
"mul_one",
"pi.smul_apply",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_succ (n : ℕ) :
diagonal k G (n + 1) ≅ left_regular k G ⊗ trivial k G ((fin n → G) →₀ k) | (linearization k G).map_iso (Action_diagonal_succ G n)
≪≫ (as_iso ((linearization k G).μ (Action.left_regular G) _)).symm
≪≫ tensor_iso (iso.refl _) (linearization_trivial_iso k G (fin n → G)) | def | group_cohomology.resolution.diagonal_succ | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Action.left_regular"
] | 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)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal_succ_hom_single (f : Gⁿ⁺¹) (a : k) :
(diagonal_succ k G n).hom.hom (single f a) =
single (f 0) 1 ⊗ₜ single (λ i, (f i.cast_succ)⁻¹ * f i.succ) a | begin
dunfold diagonal_succ,
simpa only [iso.trans_hom, iso.symm_hom, Action.comp_hom, Module.comp_def, linear_map.comp_apply,
functor.map_iso_hom, linearization_map_hom_single (Action_diagonal_succ G n).hom f a,
as_iso_inv, linearization_μ_inv_hom, Action_diagonal_succ_hom_apply, finsupp_tensor_finsupp',
... | lemma | group_cohomology.resolution.diagonal_succ_hom_single | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Action.comp_hom",
"Module.comp_def",
"finsupp_tensor_finsupp'",
"finsupp_tensor_finsupp_symm_single",
"linear_equiv.coe_to_linear_map",
"linear_equiv.trans_apply",
"linear_equiv.trans_symm",
"linear_map.comp_apply",
"tensor_product.lid_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_succ_inv_single_single (g : G) (f : Gⁿ) (a b : k) :
(diagonal_succ k G n).inv.hom (finsupp.single g a ⊗ₜ finsupp.single f b) =
single (g • partial_prod f) (a * b) | begin
dunfold diagonal_succ,
simp only [iso.trans_inv, iso.symm_inv, iso.refl_inv, tensor_iso_inv, Action.tensor_hom,
Action.comp_hom, Module.comp_def, linear_map.comp_apply, as_iso_hom, functor.map_iso_inv,
Module.monoidal_category.hom_apply, linearization_trivial_iso_inv_hom_apply,
linearization_μ_hom... | lemma | group_cohomology.resolution.diagonal_succ_inv_single_single | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Action.comp_hom",
"Action.id_hom",
"Action.left_regular",
"Action.tensor_hom",
"Module.comp_def",
"Module.id_apply",
"Module.monoidal_category.hom_apply",
"finsupp.single",
"finsupp_tensor_finsupp'_single_tmul_single",
"linear_equiv.coe_to_linear_map",
"linear_map.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_succ_inv_single_left (g : G) (f : Gⁿ →₀ k) (r : k) :
(diagonal_succ k G n).inv.hom (finsupp.single g r ⊗ₜ f) =
finsupp.lift (Gⁿ⁺¹ →₀ k) k Gⁿ (λ f, single (g • partial_prod f) r) f | begin
refine f.induction _ _,
{ simp only [tensor_product.tmul_zero, map_zero] },
{ intros a b x ha hb hx,
simp only [lift_apply, smul_single', mul_one, tensor_product.tmul_add, map_add,
diagonal_succ_inv_single_single, hx, finsupp.sum_single_index,
mul_comm b, zero_mul, single_zero] },
end | lemma | group_cohomology.resolution.diagonal_succ_inv_single_left | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"finsupp.lift",
"finsupp.single",
"mul_comm",
"mul_one",
"tensor_product.tmul_add",
"tensor_product.tmul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_succ_inv_single_right (g : G →₀ k) (f : Gⁿ) (r : k) :
(diagonal_succ k G n).inv.hom (g ⊗ₜ finsupp.single f r) =
finsupp.lift _ k G (λ a, single (a • partial_prod f) r) g | begin
refine g.induction _ _,
{ simp only [tensor_product.zero_tmul, map_zero], },
{ intros a b x ha hb hx,
simp only [lift_apply, smul_single', map_add, hx, diagonal_succ_inv_single_single,
tensor_product.add_tmul, finsupp.sum_single_index, zero_mul, single_zero] }
end | lemma | group_cohomology.resolution.diagonal_succ_inv_single_right | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"finsupp.lift",
"finsupp.single",
"tensor_product.add_tmul",
"tensor_product.zero_tmul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_action_basis_aux : (monoid_algebra k G ⊗[k] ((fin n → G) →₀ k)) ≃ₗ[monoid_algebra k G]
(of_mul_action k G (fin (n + 1) → G)).as_module | { map_smul' := λ r x,
begin
rw [ring_hom.id_apply, linear_equiv.to_fun_eq_coe, ←linear_equiv.map_smul],
congr' 1,
refine x.induction_on _ (λ x y, _) (λ y z hy hz, _),
{ simp only [smul_zero] },
{ simp only [tensor_product.smul_tmul'],
show (r * x) ⊗ₜ y = _,
rw [←of_mul_action_self_smul... | def | group_cohomology.resolution.of_mul_action_basis_aux | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"linear_equiv.to_fun_eq_coe",
"monoid_algebra",
"ring_hom.id_apply",
"smul_add",
"smul_zero",
"tensor_product.smul_tmul'"
] | The `k[G]`-linear isomorphism `k[G] ⊗ₖ k[Gⁿ] ≃ k[Gⁿ⁺¹]`, where the `k[G]`-module structure on
the lefthand side is `tensor_product.left_module`, whilst that of the righthand side comes from
`representation.as_module`. Allows us to use `basis.algebra_tensor_product` to get a `k[G]`-basis
of the righthand side. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_mul_action_basis :
basis (fin n → G) (monoid_algebra k G) (of_mul_action k G (fin (n + 1) → G)).as_module | @basis.map _ (monoid_algebra k G) (monoid_algebra k G ⊗[k] ((fin n → G) →₀ k))
_ _ _ _ _ _ (@algebra.tensor_product.basis k _ (monoid_algebra k G) _ _ ((fin n → G) →₀ k) _ _
(fin n → G) ⟨linear_equiv.refl k _⟩) (of_mul_action_basis_aux k G n) | def | group_cohomology.resolution.of_mul_action_basis | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"algebra.tensor_product.basis",
"basis",
"basis.map",
"monoid_algebra"
] | A `k[G]`-basis of `k[Gⁿ⁺¹]`, coming from the `k[G]`-linear isomorphism
`k[G] ⊗ₖ k[Gⁿ] ≃ k[Gⁿ⁺¹].` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_mul_action_free :
module.free (monoid_algebra k G) (of_mul_action k G (fin (n + 1) → G)).as_module | module.free.of_basis (of_mul_action_basis k G n) | lemma | group_cohomology.resolution.of_mul_action_free | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"module.free",
"module.free.of_basis",
"monoid_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_hom_equiv :
(Rep.of_mul_action k G (fin (n + 1) → G) ⟶ A) ≃ₗ[k] ((fin n → G) → A) | linear.hom_congr k ((diagonal_succ k G n).trans
((representation.of_mul_action k G G).Rep_of_tprod_iso 1)) (iso.refl _) ≪≫ₗ
((Rep.monoidal_closed.linear_hom_equiv_comm _ _ _) ≪≫ₗ (Rep.left_regular_hom_equiv _))
≪≫ₗ (finsupp.llift A k k (fin n → G)).symm | def | Rep.diagonal_hom_equiv | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Rep.left_regular_hom_equiv",
"Rep.monoidal_closed.linear_hom_equiv_comm",
"Rep.of_mul_action",
"finsupp.llift",
"representation.of_mul_action"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal_hom_equiv_apply (f : Rep.of_mul_action k G (fin (n + 1) → G) ⟶ A)
(x : fin n → G) : diagonal_hom_equiv n A f x = f.hom (finsupp.single (fin.partial_prod x) 1) | begin
unfold diagonal_hom_equiv,
simpa only [linear_equiv.trans_apply, Rep.left_regular_hom_equiv_apply,
monoidal_closed.linear_hom_equiv_comm_hom, finsupp.llift_symm_apply, tensor_product.curry_apply,
linear.hom_congr_apply, iso.refl_hom, iso.trans_inv, Action.comp_hom, Module.comp_def,
linear_map.comp... | lemma | Rep.diagonal_hom_equiv_apply | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Action.comp_hom",
"Module.comp_def",
"Rep.of_mul_action",
"fin.partial_prod",
"finsupp.llift_symm_apply",
"finsupp.single",
"linear_equiv.trans_apply",
"linear_map.comp_apply",
"one_mul",
"one_smul",
"representation.Rep_of_tprod_iso_inv_apply",
"tensor_product.curry_apply"
] | Given a `k`-linear `G`-representation `A`, `diagonal_hom_equiv` 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).` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal_hom_equiv_symm_apply (f : (fin n → G) → A) (x : fin (n + 1) → G) :
((diagonal_hom_equiv n A).symm f).hom (finsupp.single x 1)
= A.ρ (x 0) (f (λ (i : fin n), (x i.cast_succ)⁻¹ * x i.succ)) | begin
unfold diagonal_hom_equiv,
simp only [linear_equiv.trans_symm, linear_equiv.symm_symm, linear_equiv.trans_apply,
Rep.left_regular_hom_equiv_symm_apply, linear.hom_congr_symm_apply, Action.comp_hom,
iso.refl_inv, category.comp_id, Rep.monoidal_closed.linear_hom_equiv_comm_symm_hom,
iso.trans_hom, M... | lemma | Rep.diagonal_hom_equiv_symm_apply | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Action.comp_hom",
"Module.comp_def",
"Rep.Action_ρ_eq_ρ",
"Rep.ihom_obj_ρ_apply",
"Rep.monoidal_closed.linear_hom_equiv_comm_symm_hom",
"Rep.of_ρ",
"Rep.trivial_def",
"finsupp.lift_apply",
"finsupp.llift_apply",
"finsupp.single",
"linear_equiv.symm_symm",
"linear_equiv.trans_apply",
"linear... | Given a `k`-linear `G`-representation `A`, `diagonal_hom_equiv` 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, ..., g... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal_hom_equiv_symm_partial_prod_succ
(f : (fin n → G) → A) (g : fin (n + 1) → G) (a : fin (n + 1)) :
((diagonal_hom_equiv n A).symm f).hom (finsupp.single (fin.partial_prod g ∘ a.succ.succ_above) 1)
= f (fin.contract_nth a (*) g) | begin
simp only [diagonal_hom_equiv_symm_apply, function.comp_app, fin.succ_succ_above_zero,
fin.partial_prod_zero, map_one, fin.succ_succ_above_succ,
linear_map.one_apply, fin.partial_prod_succ],
congr,
ext,
rw [←fin.partial_prod_succ, fin.inv_partial_prod_mul_eq_contract_nth],
end | lemma | Rep.diagonal_hom_equiv_symm_partial_prod_succ | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"fin.contract_nth",
"fin.inv_partial_prod_mul_eq_contract_nth",
"fin.partial_prod",
"fin.partial_prod_succ",
"fin.partial_prod_zero",
"fin.succ_succ_above_succ",
"fin.succ_succ_above_zero",
"finsupp.single",
"linear_map.one_apply",
"map_one"
] | Auxiliary lemma for defining group cohomology, used to show that the isomorphism
`diagonal_hom_equiv` commutes with the differentials in two complexes which compute
group cohomology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
classifying_space_universal_cover [monoid G] :
simplicial_object (Action (Type u) $ Mon.of G) | { obj := λ n, Action.of_mul_action G (fin (n.unop.len + 1) → G),
map := λ m n f,
{ hom := λ x, x ∘ f.unop.to_order_hom,
comm' := λ g, rfl },
map_id' := λ n, rfl,
map_comp' := λ i j k f g, rfl } | def | classifying_space_universal_cover | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Action",
"Action.of_mul_action",
"Mon.of",
"monoid"
] | The simplicial `G`-set sending `[n]` to `Gⁿ⁺¹` equipped with the diagonal action of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cech_nerve_terminal_from_iso :
cech_nerve_terminal_from (Action.of_mul_action G G) ≅ classifying_space_universal_cover G | nat_iso.of_components (λ n, limit.iso_limit_cone (Action.of_mul_action_limit_cone _ _)) $ λ m n f,
begin
refine is_limit.hom_ext (Action.of_mul_action_limit_cone.{u 0} _ _).2 (λ j, _),
dunfold cech_nerve_terminal_from pi.lift,
dsimp,
rw [category.assoc, limit.iso_limit_cone_hom_π, limit.lift_π, category.assoc],... | def | classifying_space_universal_cover.cech_nerve_terminal_from_iso | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Action.of_mul_action",
"Action.of_mul_action_limit_cone",
"classifying_space_universal_cover"
] | When the category is `G`-Set, `cech_nerve_terminal_from` of `G` with the left regular action is
isomorphic to `EG`, the universal cover of the classifying space of `G` as a simplicial `G`-set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cech_nerve_terminal_from_iso_comp_forget :
cech_nerve_terminal_from G ≅ classifying_space_universal_cover G ⋙ forget _ | nat_iso.of_components (λ n, types.product_iso _) $ λ m n f, matrix.ext $ λ i j,
types.limit.lift_π_apply _ _ _ _ | def | classifying_space_universal_cover.cech_nerve_terminal_from_iso_comp_forget | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"classifying_space_universal_cover",
"matrix.ext"
] | As a simplicial set, `cech_nerve_terminal_from` of a monoid `G` is isomorphic to the universal
cover of the classifying space of `G` as a simplicial set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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