mathlib documentation

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.

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noncomputable def Rep.abelian (k G : Type u) [ring k] [monoid G] :
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def Rep (k G : Type u) [ring k] [monoid G] :
Type (u+1)

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

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def Rep.add_comm_group {k G : Type u} [comm_ring k] [monoid G] (V : Rep k G) :
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def Rep.module {k G : Type u} [comm_ring k] [monoid G] (V : Rep k G) :
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def Rep.ρ {k G : Type u} [comm_ring k] [monoid G] (V : Rep k G) :

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

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

Lift an unbundled representation to Rep.

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theorem Rep.coe_of {k G : Type u} [comm_ring k] [monoid G] {V : Type u} [add_comm_group V] [module k V] (ρ : G →* V →ₗ[k] V) :
(Rep.of ρ) = V
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theorem Rep.of_ρ {k G : Type u} [comm_ring k] [monoid G] {V : Type u} [add_comm_group V] [module k V] (ρ : G →* V →ₗ[k] V) :
(Rep.of ρ).ρ = ρ
noncomputable def Rep.linearization (k G : Type u) [comm_ring k] [monoid G] :

The monoidal functor sending a type H with a G-action to the induced k-linear G-representation on k[H].

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theorem Rep.linearization_obj_ρ {k G : Type u} [comm_ring k] [monoid G] (X : Action (Type u) (Mon.of G)) (g : G) (x : X.V →₀ k) :
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theorem Rep.linearization_of {k G : Type u} [comm_ring k] [monoid G] (X : Action (Type u) (Mon.of G)) (g : G) (x : X.V) :
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noncomputable def Rep.of_mul_action (k G : Type u) [comm_ring k] [monoid G] (H : Type u) [mul_action G H] :
Rep k G

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.

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.

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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) [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.lift k G (V →ₗ[k] V)) ρ) r) x) = (((monoid_algebra.lift k G (W →ₗ[k] W)) σ) r) (f x)

Auxilliary lemma for to_Module_monoid_algebra.

Auxilliary definition for to_Module_monoid_algebra.

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

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

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

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

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Auxilliary definition for equivalence_Module_monoid_algebra.

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Auxilliary definition for equivalence_Module_monoid_algebra.

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Auxilliary definition for equivalence_Module_monoid_algebra.

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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.

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

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

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