Omega-equivariance of the symmetric-group action

The braiding-generated symmetric-group action on the skein endomorphism algebra skeinEnd f n transports along the Deligne package's fibre functor ω to a well-defined algebra homomorphism SymGroupAlgebra n →ₐ[ℂ] End (ω.obj (SkeinObj.mk n)).

Main results

• omegaPermHom -- the monoid homomorphism Perm (Fin n) →* End (ω.obj (SkeinObj.mk n)) obtained by composing the permutation-to-endomorphism map with the functorial action on endomorphisms.
• omegaSkeinRep -- the algebra homomorphism SymGroupAlgebra n →ₐ[ℂ] End (ω.obj (SkeinObj.mk n)) lifted from omegaPermHom via the universal property of the group algebra.
• omegaSkeinRep_of -- on a single permutation σ, the representation yields ω.map (permClass f n σ).
• omegaSkeinRep_eq -- the transported representation agrees with applying ω.map to the skein representation: omegaSkeinRep f P n x = ω.map (skeinRep f n x).

Formulation

The skein category's symmetric-group action is the algebra homomorphism skeinRep f n : SymGroupAlgebra n →ₐ[ℂ] skeinEnd f n built from σ ↦ [permFragment σ] (see RS.Novel.Envelope.SkeinTower). The fibre functor ω from a Deligne package induces a ring homomorphism on endomorphisms via functoriality. The composite ω.map ∘ skeinRep f n is therefore an algebra homomorphism from the symmetric-group algebra to End (ω.obj (SkeinObj.mk n)), and agreeing on the generators σ makes it that composite.

The transported permutation representation

noncomputable def RS.omegaPermHom {R : ℕ} (f : EdgeRankParameter R) (P : DelignePackage (SkeinObj f)) (n : ℕ) : Equiv.Perm (Fin n) →* CategoryTheory.End (P.ω.obj { arity := n })

The monoid homomorphism sending a permutation σ : Perm (Fin n) to the endomorphism ω.map (permClass f n σ) of the image object. This is the composition of the skein permutation-to-endomorphism map permToEnd f n with the functorial action ω.mapEnd.

Equations

• RS.omegaPermHom f P n = (CategoryTheory.Functor.mapEnd { arity := n } P.ω).comp (RS.permToEnd f n)

noncomputable def RS.omegaSkeinRep {R : ℕ} (f : EdgeRankParameter R) (P : DelignePackage (SkeinObj f)) (n : ℕ) : SymGroupAlgebra n →ₐ[ℂ] CategoryTheory.End (P.ω.obj { arity := n })

The transported symmetric-group representation: the algebra homomorphism SymGroupAlgebra n →ₐ[ℂ] End (ω.obj (SkeinObj.mk n)) obtained by lifting omegaPermHom through the universal property of the group algebra.

Equations

• RS.omegaSkeinRep f P n = (MonoidAlgebra.lift ℂ (CategoryTheory.End (P.ω.obj { arity := n })) (Equiv.Perm (Fin n))) (RS.omegaPermHom f P n)

theorem RS.omegaSkeinRep_of {R : ℕ} (f : EdgeRankParameter R) (P : DelignePackage (SkeinObj f)) (n : ℕ) (σ : Equiv.Perm (Fin n)) : (omegaSkeinRep f P n) ((MonoidAlgebra.of ℂ (Equiv.Perm (Fin n))) σ) = P.ω.map (permClass f n σ)

On a single permutation, the transported representation yields ω.map (permClass f n σ).

theorem RS.omegaSkeinRep_eq {R : ℕ} (f : EdgeRankParameter R) (P : DelignePackage (SkeinObj f)) (n : ℕ) (x : SymGroupAlgebra n) : (omegaSkeinRep f P n) x = P.ω.map ((skeinRep f n) x)

Equivariance: the transported representation agrees with applying ω.map to the skein representation element by element. Both sides are algebra homs agreeing on generators, hence equal on all elements by the universal property.