# mathlibdocumentation

representation_theory.Rep

# Rep k G is the category of k-linear representations of G. #

If V : Rep k G, there is a coercion that allows you to treat V as a type, and this type comes equipped with a module k V instance. Also V.ρ gives the homomorphism G →* (V →ₗ[k] V).

Conversely, given a homomorphism ρ : G →* (V →ₗ[k] V), you can construct the bundled representation as Rep.of ρ.

We construct the categorical equivalence Rep k G ≌ Module (monoid_algebra k G). We verify that Rep k G is a k-linear abelian symmetric monoidal category with all (co)limits.

@[protected, instance]
def Rep.has_colimits (k G : Type u) [ring k] [monoid G] :
@[protected, instance]
def Rep.preadditive (k G : Type u) [ring k] [monoid G] :
@[protected, instance]
noncomputable def Rep.abelian (k G : Type u) [ring k] [monoid G] :
@[protected, instance]
def Rep.large_category (k G : Type u) [ring k] [monoid G] :
@[protected, instance]
def Rep.has_limits (k G : Type u) [ring k] [monoid G] :
@[reducible]
def Rep (k G : Type u) [ring k] [monoid G] :
Type (u+1)

The category of k-linear representations of a monoid G.

@[protected, instance]
def Rep.concrete_category (k G : Type u) [ring k] [monoid G] :
@[protected, instance]
def Rep.category_theory.linear (k G : Type u) [comm_ring k] [monoid G] :
(Rep k G)
Equations
@[protected, instance]
def Rep.has_coe_to_sort {k G : Type u} [comm_ring k] [monoid G] :
has_coe_to_sort (Rep k G) (Type u)
Equations
@[protected, instance]
def Rep.add_comm_group {k G : Type u} [comm_ring k] [monoid G] (V : Rep k G) :
Equations
@[protected, instance]
def Rep.module {k G : Type u} [comm_ring k] [monoid G] (V : Rep k G) :
V
Equations
def Rep.ρ {k G : Type u} [comm_ring k] [monoid G] (V : Rep k G) :
V

Specialize the existing Action.ρ, changing the type to representation k G V.

Equations
def Rep.of {k G : Type u} [comm_ring k] [monoid G] {V : Type u} [ V] (ρ : G →* V →ₗ[k] V) :
Rep k G

Lift an unbundled representation to Rep.

Equations
• = {V := V _inst_4, ρ := ρ}
@[simp]
theorem Rep.coe_of {k G : Type u} [comm_ring k] [monoid G] {V : Type u} [ V] (ρ : G →* V →ₗ[k] V) :
(Rep.of ρ) = V
@[simp]
theorem Rep.of_ρ {k G : Type u} [comm_ring k] [monoid G] {V : Type u} [ V] (ρ : G →* V →ₗ[k] V) :
(Rep.of ρ).ρ = ρ

# The categorical equivalence Rep k G ≌ Module.{u} (monoid_algebra k G). #

theorem Rep.to_Module_monoid_algebra_map_aux {k : Type u_1} {G : Type u_2} [comm_ring k] [monoid G] (V : Type u_3) (W : Type u_4) [ V] [ 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 : G) (x : V) :
f ((( (V →ₗ[k] V)) ρ) r) x) = (( (W →ₗ[k] W)) σ) r) (f x)

Auxilliary lemma for to_Module_monoid_algebra.

def Rep.to_Module_monoid_algebra_map {k G : Type u} [comm_ring k] [monoid G] {V W : Rep k G} (f : V W) :

Auxilliary definition for to_Module_monoid_algebra.

Equations
noncomputable def Rep.to_Module_monoid_algebra {k G : Type u} [comm_ring k] [monoid G] :
Rep k G Module G)

Functorially convert a representation of G into a module over monoid_algebra k G.

Equations
noncomputable def Rep.of_Module_monoid_algebra {k G : Type u} [comm_ring k] [monoid G] :
Module G) Rep k G

Functorially convert a module over monoid_algebra k G into a representation of G.

Equations
theorem Rep.of_Module_monoid_algebra_obj_coe {k G : Type u} [comm_ring k] [monoid G] (M : Module G)) :
= G) M
theorem Rep.of_Module_monoid_algebra_obj_ρ {k G : Type u} [comm_ring k] [monoid G] (M : Module G)) :
noncomputable def Rep.counit_iso_add_equiv {k G : Type u} [comm_ring k] [monoid G] {M : Module G)} :

Auxilliary definition for equivalence_Module_monoid_algebra.

Equations
noncomputable def Rep.unit_iso_add_equiv {k G : Type u} [comm_ring k] [monoid G] {V : Rep k G} :

Auxilliary definition for equivalence_Module_monoid_algebra.

Equations
noncomputable def Rep.counit_iso {k G : Type u} [comm_ring k] [monoid G] (M : Module G)) :

Auxilliary definition for equivalence_Module_monoid_algebra.

Equations
theorem Rep.unit_iso_comm {k G : Type u} [comm_ring k] [monoid G] (V : Rep k G) (g : G) (x : V) :
noncomputable def Rep.unit_iso {k G : Type u} [comm_ring k] [monoid G] (V : Rep k G) :

Auxilliary definition for equivalence_Module_monoid_algebra.

Equations
noncomputable def Rep.equivalence_Module_monoid_algebra {k G : Type u} [comm_ring k] [monoid G] :
Rep k G Module G)

The categorical equivalence Rep k G ≌ Module (monoid_algebra k G).

Equations