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Mathlib.RepresentationTheory.Basic

Monoid representations #

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

Main definitions #

Implementation notes #

Representations of a monoid G on a k-module V are implemented as homomorphisms G →* (V →ₗ[k] V). We use the abbreviation Representation for this hom space.

The theorem asAlgebraHom_def constructs a module over the group k-algebra of G (implemented as MonoidAlgebra k G) corresponding to a representation. If ρ : Representation k G V, this module can be accessed via ρ.asModule. Conversely, given a MonoidAlgebra k G-module M M.ofModule` is the associociated representation seen as a homomorphism.

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

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

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    def Representation.trivial (k : Type u_1) {G : Type u_2} {V : Type u_3} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] :

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

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      theorem Representation.trivial_def (k : Type u_1) {G : Type u_2} {V : Type u_3} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] (g : G) (v : V) :
      class Representation.IsTrivial {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] (ρ : Representation k G V) :

      A predicate for representations that fix every element.

      • out : ∀ (g : G) (x : V), (ρ g) x = x
      Instances
        @[simp]
        theorem Representation.apply_eq_self {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] (ρ : Representation k G V) (g : G) (x : V) [h : Representation.IsTrivial ρ] :
        (ρ g) x = x
        noncomputable def Representation.asAlgebraHom {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] (ρ : Representation k G V) :

        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.

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          @[simp]
          theorem Representation.asAlgebraHom_single {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] (ρ : Representation k G V) (g : G) (r : k) :
          theorem Representation.asAlgebraHom_single_one {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] (ρ : Representation k G V) (g : G) :
          theorem Representation.asAlgebraHom_of {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] (ρ : Representation k G V) (g : G) :
          def Representation.asModule {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] :
          Representation k G VType 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.

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            noncomputable instance Representation.asModuleModule {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] (ρ : Representation k G V) :

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

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

            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.

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              @[simp]
              noncomputable def Representation.ofModule' {k : Type u_1} {G : Type u_2} [CommSemiring k] [Monoid G] (M : Type u_4) [AddCommMonoid M] [Module k M] [Module (MonoidAlgebra k G) M] [IsScalarTower k (MonoidAlgebra k G) 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.)

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                noncomputable def Representation.ofModule {k : Type u_1} {G : Type u_2} [CommSemiring k] [Monoid G] (M : Type u_4) [AddCommMonoid M] [Module (MonoidAlgebra k G) 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.

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                  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).

                  noncomputable def Representation.ofMulAction (k : Type u_1) [CommSemiring k] (G : Type u_2) [Monoid G] (H : Type u_3) [MulAction G H] :

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

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                    theorem Representation.ofMulAction_def {k : Type u_1} [CommSemiring k] {G : Type u_2} [Monoid G] {H : Type u_3} [MulAction G H] (g : G) :
                    (Representation.ofMulAction k G H) g = Finsupp.lmapDomain k k fun (x : H) => g x
                    theorem Representation.ofMulAction_single {k : Type u_1} [CommSemiring k] {G : Type u_2} [Monoid G] {H : Type u_3} [MulAction G H] (g : G) (x : H) (r : k) :

                    Turns a k-module A with a compatible DistribMulAction of a monoid G into a k-linear G-representation on A.

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                    • One or more equations did not get rendered due to their size.
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                      @[simp]
                      theorem Representation.ofDistribMulAction_apply_apply {k : Type u_1} {G : Type u_2} {A : Type u_3} [CommSemiring k] [Monoid G] [AddCommMonoid A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] (g : G) (a : A) :
                      @[simp]
                      theorem Representation.ofMulDistribMulAction_apply_apply (M : Type u_1) (G : Type u_2) [Monoid M] [CommGroup G] [MulDistribMulAction M G] (g : M) (a : Additive G) :
                      ((Representation.ofMulDistribMulAction M G) g) a = Additive.ofMul (g Additive.toMul a)
                      @[simp]
                      theorem Representation.ofMulAction_apply {k : Type u_1} {G : Type u_2} [CommSemiring k] [Group G] {H : Type u_4} [MulAction G H] (g : G) (f : H →₀ k) (h : H) :
                      (((Representation.ofMulAction k G H) g) f) h = f (g⁻¹ h)
                      @[simp]
                      theorem Representation.ofMulActionSelfAsModuleEquiv_symm_apply {k : Type u_1} {G : Type u_2} [CommSemiring k] [Group G] :
                      ∀ (a : G →₀ k), (LinearEquiv.symm Representation.ofMulActionSelfAsModuleEquiv) a = (Representation.asModuleEquiv (Representation.ofMulAction k G G)).invFun a
                      @[simp]
                      theorem Representation.ofMulActionSelfAsModuleEquiv_apply {k : Type u_1} {G : Type u_2} [CommSemiring k] [Group G] :
                      ∀ (a : Representation.asModule (Representation.ofMulAction k G G)), Representation.ofMulActionSelfAsModuleEquiv a = (Representation.asModuleEquiv (Representation.ofMulAction k G G)).toFun a

                      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.

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                      • One or more equations did not get rendered due to their size.
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                        def Representation.asGroupHom {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Group G] [AddCommMonoid V] [Module k V] (ρ : Representation k G V) :
                        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.

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                          theorem Representation.asGroupHom_apply {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Group G] [AddCommMonoid V] [Module k V] (ρ : Representation k G V) (g : G) :
                          ((Representation.asGroupHom ρ) g) = ρ g
                          noncomputable def Representation.tprod {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] [AddCommMonoid W] [Module k W] (ρV : Representation k G V) (ρW : Representation k G W) :

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

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                            @[simp]
                            theorem Representation.tprod_apply {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] [AddCommMonoid W] [Module k W] (ρV : Representation k G V) (ρW : Representation k G W) (g : G) :
                            (Representation.tprod ρV ρW) 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} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] [AddCommMonoid W] [Module k W] (ρV : Representation k G V) (r : MonoidAlgebra k G) (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} [CommSemiring k] [Monoid G] [AddCommMonoid V] [Module k V] [AddCommMonoid W] [Module k W] (ρW : Representation k G W) (r : MonoidAlgebra k G) (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} [CommSemiring k] [Group G] [AddCommMonoid V] [Module k V] [AddCommMonoid W] [Module k W] (ρV : Representation k G V) (ρW : Representation k G W) :

                            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.

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                            • One or more equations did not get rendered due to their size.
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                              @[simp]
                              theorem Representation.linHom_apply {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [CommSemiring k] [Group G] [AddCommMonoid V] [Module k V] [AddCommMonoid W] [Module k W] (ρV : Representation k G V) (ρW : Representation k G W) (g : G) (f : V →ₗ[k] W) :
                              ((Representation.linHom ρV ρW) g) f = ρW g ∘ₗ f ∘ₗ ρV g⁻¹
                              def Representation.dual {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Group G] [AddCommMonoid V] [Module k V] (ρV : Representation k G V) :

                              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.

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                                @[simp]
                                theorem Representation.dual_apply {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommSemiring k] [Group G] [AddCommMonoid V] [Module k V] (ρV : Representation k G V) (g : G) :
                                (Representation.dual ρV) 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} [CommSemiring k] [Group G] [AddCommMonoid V] [Module k V] [AddCommMonoid W] [Module k W] (ρV : Representation k G V) (ρW : Representation k G W) (g : G) :
                                dualTensorHom k V W ∘ₗ TensorProduct.map ((Representation.dual ρV) g) (ρW g) = (Representation.linHom ρV ρW) g ∘ₗ dualTensorHom k V W

                                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.