# Documentation

Mathlib.RepresentationTheory.Basic

# Monoid representations #

This file introduces monoid representations and their characters and defines a few ways to construct representations.

## Main definitions #

• Representation.Representation
• Representation.character
• Representation.tprod
• Representation.linHom
• Representation.dual

## Implementation notes #

Representations of a monoid G on a k-module V are implemented as homomorphisms G →* (V →ₗ[k] V).

@[inline, reducible]
abbrev Representation (k : Type u_1) (G : Type u_2) (V : Type u_3) [] [] [] [Module k V] :
Type (max u_3 u_2)

A representation of G on the k-module V is a homomorphism G →* (V →ₗ[k] V).

Instances For
def Representation.trivial (k : Type u_1) {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] :

The trivial representation of G on a k-module V.

Instances For
@[simp]
theorem Representation.trivial_def (k : Type u_1) {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (g : G) (v : V) :
↑(↑() g) v = v
noncomputable def Representation.asAlgebraHom {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) :

A k-linear representation of G on V can be thought of as an algebra map from MonoidAlgebra k G into the k-linear endomorphisms of V.

Instances For
theorem Representation.asAlgebraHom_def {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) :
= ↑(MonoidAlgebra.lift k G (V →ₗ[k] V)) ρ
@[simp]
theorem Representation.asAlgebraHom_single {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) (g : G) (r : k) :
(↑() fun₀ | g => r) = r ρ g
theorem Representation.asAlgebraHom_single_one {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) (g : G) :
(↑() fun₀ | g => 1) = ρ g
theorem Representation.asAlgebraHom_of {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) (g : G) :
↑() (↑() g) = ρ g
def Representation.asModule {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] :
Type u_3

If ρ : Representation k G V, then ρ.asModule is a type synonym for V, which we equip with an instance Module (MonoidAlgebra k G) ρ.asModule.

You should use asModuleEquiv : ρ.asModule ≃+ V to translate terms.

Instances For
instance Representation.instAddCommMonoidAsModule {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) :
instance Representation.instInhabitedAsModule {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) :
noncomputable instance Representation.asModuleModule {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) :
Module () ()

A k-linear representation of G on V can be thought of as a module over MonoidAlgebra k G.

instance Representation.instModuleAsModuleToSemiringInstAddCommMonoidAsModule {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) :
def Representation.asModuleEquiv {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) :

The additive equivalence from the Module (MonoidAlgebra k G) to the original vector space of the representative.

This is just the identity, but it is helpful for typechecking and keeping track of instances.

Instances For
@[simp]
theorem Representation.asModuleEquiv_map_smul {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) (r : ) (x : ) :
↑() (r x) = ↑() ()
@[simp]
theorem Representation.asModuleEquiv_symm_map_smul {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) (r : k) (x : V) :
↑() (r x) = ↑(algebraMap k ()) r
@[simp]
theorem Representation.asModuleEquiv_symm_map_rho {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) (g : G) (x : V) :
↑() (↑(ρ g) x) = ↑() g
noncomputable def Representation.ofModule' {k : Type u_1} {G : Type u_2} [] [] (M : Type u_4) [] [Module k M] [Module () M] [IsScalarTower k () M] :

Build a Representation k G M from a [Module (MonoidAlgebra k G) M].

This version is not always what we want, as it relies on an existing [Module k M] instance, along with a [IsScalarTower k (MonoidAlgebra k G) M] instance.

We remedy this below in ofModule (with the tradeoff that the representation is defined only on a type synonym of the original module.)

Instances For
noncomputable def Representation.ofModule {k : Type u_1} {G : Type u_2} [] [] (M : Type u_4) [] [Module () M] :

Build a Representation from a [Module (MonoidAlgebra k G) M].

Note that the representation is built on restrictScalars k (MonoidAlgebra k G) M, rather than on M itself.

Instances For

## ofModule and asModule are inverses. #

This requires a little care in both directions: this is a categorical equivalence, not an isomorphism.

See Rep.equivalenceModuleMonoidAlgebra for the full statement.

Starting with ρ : Representation k G V, converting to a module and back again we have a Representation k G (restrictScalars k (MonoidAlgebra k G) ρ.asModule). To compare these, we use the composition of restrictScalarsAddEquiv and ρ.asModuleEquiv.

Similarly, starting with Module (MonoidAlgebra k G) M, after we convert to a representation and back to a module, we have Module (MonoidAlgebra k G) (restrictScalars k (MonoidAlgebra k G) M).

@[simp]
theorem Representation.ofModule_asAlgebraHom_apply_apply {k : Type u_1} {G : Type u_2} [] [] (M : Type u_4) [] [Module () M] (r : ) (m : RestrictScalars k () M) :
↑() m = ↑(AddEquiv.symm (RestrictScalars.addEquiv k () ((fun x => M) m))) (r ↑() m)
@[simp]
theorem Representation.ofModule_asModule_act {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) (g : G) (x : ) :
↑() x = ↑(AddEquiv.symm (RestrictScalars.addEquiv k () ((fun x => ) (↑(ρ g) (↑() (↑() x)))))) (↑() (↑(ρ g) (↑() (↑() x))))
theorem Representation.smul_ofModule_asModule {k : Type u_1} {G : Type u_2} [] [] (M : Type u_4) [] [Module () M] (r : ) :
↑() (↑() (r m)) = r ↑() ()
instance Representation.instAddCommGroupAsModuleToCommSemiringToAddCommMonoid {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [I : ] [Module k V] (ρ : ) :
noncomputable def Representation.ofMulAction (k : Type u_1) [] (G : Type u_2) [] (H : Type u_3) [] :

A G-action on H induces a representation G →* End(k[H]) in the natural way.

Instances For
theorem Representation.ofMulAction_def {k : Type u_1} [] {G : Type u_2} [] {H : Type u_3} [] (g : G) :
↑() g = Finsupp.lmapDomain k k ((fun x x_1 => x x_1) g)
theorem Representation.ofMulAction_single {k : Type u_1} [] {G : Type u_2} [] {H : Type u_3} [] (g : G) (x : H) (r : k) :
(↑(↑() g) fun₀ | x => r) = fun₀ | g x => r
@[simp]
theorem Representation.ofMulAction_apply {k : Type u_1} {G : Type u_2} [] [] {H : Type u_4} [] (g : G) (f : H →₀ k) (h : H) :
↑(↑(↑() g) f) h = f (g⁻¹ h)
theorem Representation.ofMulAction_self_smul_eq_mul {k : Type u_1} {G : Type u_2} [] [] (x : ) (y : ) :
x y = x * y
@[simp]
theorem Representation.ofMulActionSelfAsModuleEquiv_apply {k : Type u_1} {G : Type u_2} [] [] :
∀ (a : ), Representation.ofMulActionSelfAsModuleEquiv a = Equiv.toFun ().toEquiv a
@[simp]
theorem Representation.ofMulActionSelfAsModuleEquiv_symm_apply {k : Type u_1} {G : Type u_2} [] [] :
∀ (a : G →₀ k), ↑(LinearEquiv.symm Representation.ofMulActionSelfAsModuleEquiv) a = Equiv.invFun ().toEquiv a
noncomputable def Representation.ofMulActionSelfAsModuleEquiv {k : Type u_1} {G : Type u_2} [] [] :

If we equip k[G] with the k-linear G-representation induced by the left regular action of G on itself, the resulting object is isomorphic as a k[G]-module to k[G] with its natural k[G]-module structure.

Instances For
def Representation.asGroupHom {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) :
G →* (V →ₗ[k] V)ˣ

When G is a group, a k-linear representation of G on V can be thought of as a group homomorphism from G into the invertible k-linear endomorphisms of V.

Instances For
theorem Representation.asGroupHom_apply {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρ : ) (g : G) :
↑() = ρ g
def Representation.tprod {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [] [] [] [Module k V] [] [Module k W] (ρV : ) (ρW : ) :

Given representations of G on V and W, there is a natural representation of G on their tensor product V ⊗[k] W.

Instances For
@[simp]
theorem Representation.tprod_apply {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [] [] [] [Module k V] [] [Module k W] (ρV : ) (ρW : ) (g : G) :
↑() g = TensorProduct.map (ρV g) (ρW g)
theorem Representation.smul_tprod_one_asModule {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [] [] [] [Module k V] [] [Module k W] (ρV : ) (r : ) (x : V) (y : W) :
let x' := x; let z := x ⊗ₜ[k] y; r z = (r x') ⊗ₜ[k] y
theorem Representation.smul_one_tprod_asModule {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [] [] [] [Module k V] [] [Module k W] (ρW : ) (r : ) (x : V) (y : W) :
let y' := y; let z := x ⊗ₜ[k] y; r z = x ⊗ₜ[k] (r y')
def Representation.linHom {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [] [] [] [Module k V] [] [Module k W] (ρV : ) (ρW : ) :

Given representations of G on V and W, there is a natural representation of G on the module V →ₗ[k] W, where G acts by conjugation.

Instances For
@[simp]
theorem Representation.linHom_apply {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [] [] [] [Module k V] [] [Module k W] (ρV : ) (ρW : ) (g : G) (f : V →ₗ[k] W) :
↑(↑() g) f = LinearMap.comp (ρW g) (LinearMap.comp f (ρV g⁻¹))
def Representation.dual {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρV : ) :

The dual of a representation ρ of G on a module V, given by (dual ρ) g f = f ∘ₗ (ρ g⁻¹), where f : Module.Dual k V.

Instances For
@[simp]
theorem Representation.dual_apply {k : Type u_1} {G : Type u_2} {V : Type u_3} [] [] [] [Module k V] (ρV : ) (g : G) :
↑() g = Module.Dual.transpose (ρV g⁻¹)
theorem Representation.dualTensorHom_comm {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [] [] [] [Module k V] [] [Module k W] (ρV : ) (ρW : ) (g : G) :
LinearMap.comp () (TensorProduct.map (↑() g) (ρW g)) = LinearMap.comp (↑() g) ()

Given $k$-modules $V, W$, there is a homomorphism $φ : V^* ⊗ W → Hom_k(V, W)$ (implemented by LinearAlgebra.Contraction.dualTensorHom). Given representations of $G$ on $V$ and $W$,there are representations of $G$ on $V^* ⊗ W$ and on $Hom_k(V, W)$. This lemma says that $φ$ is $G$-linear.