SchurWeyl duality over arbitrary commutative rings Tiago Cruz - PowerPoint PPT Presentation

Schur–Weyl duality over arbitrary commutative rings Tiago Cruz 11/06/2019 Spa, Belgium

Classical Schur–Weyl duality V = R n GL n ( R ) - the general linear group of degree n over R S d - the symmetric group on d letters 2

Classical Schur–Weyl duality V = R n GL n ( R ) - the general linear group of degree n over R S d - the symmetric group on d letters GL n ( R ) V ⊗ d S d GL n ( R ) ∋ g : g ( v 1 ⊗ · · · ⊗ v d ) = gv 1 ⊗ · · · ⊗ gv d S d ∋ σ : ( v 1 ⊗ · · · ⊗ v d ) σ = v σ (1) ⊗ · · · ⊗ v σ ( d ) 2

Classical Schur–Weyl duality V = R n GL n ( R ) - the general linear group of degree n over R S d - the symmetric group on d letters V ⊗ d GL n ( R ) S d GL n ( R ) ∋ g : g ( v 1 ⊗ · · · ⊗ v d ) = gv 1 ⊗ · · · ⊗ gv d S d ∋ σ : ( v 1 ⊗ · · · ⊗ v d ) σ = v σ (1) ⊗ · · · ⊗ v σ ( d ) Extending the actions to the group algebras we obtain the algebra homomorphisms � V ⊗ d � ρ : RGL n ( R ) → End R � V ⊗ d � ψ : RS d → End R . 2

Classical Schur–Weyl duality V = R n GL n ( R ) - the general linear group of degree n over R S d - the symmetric group on d letters GL n ( R ) V ⊗ d S d GL n ( R ) ∋ g : g ( v 1 ⊗ · · · ⊗ v d ) = gv 1 ⊗ · · · ⊗ gv d S d ∋ σ : ( v 1 ⊗ · · · ⊗ v d ) σ = v σ (1) ⊗ · · · ⊗ v σ ( d ) Extending the actions to the group algebras we obtain the algebra homomorphisms � V ⊗ d � ρ : RGL n ( R ) → End RS d � V ⊗ d � ψ : RS d → End RGL n ( R ) . 2

Classical Schur–Weyl duality V = R n GL n ( R ) - the general linear group of degree n over R S d - the symmetric group on d letters GL n ( R ) V ⊗ d S d GL n ( R ) ∋ g : g ( v 1 ⊗ · · · ⊗ v d ) = gv 1 ⊗ · · · ⊗ gv d S d ∋ σ : ( v 1 ⊗ · · · ⊗ v d ) σ = v σ (1) ⊗ · · · ⊗ v σ ( d ) Extending the actions to the group algebras we obtain the algebra homomorphisms � V ⊗ d � ρ : RGL n ( R ) → End RS d � V ⊗ d � ψ : RS d → End RGL n ( R ) . � V ⊗ d � The endomorphism algebra S R ( n , d ) = End RS d is called Schur algebra . 2

Classical Schur–Weyl duality GL n ( R ) V ⊗ d S d GL n ( R ) ∋ g : g ( v 1 ⊗ · · · ⊗ v d ) = gv 1 ⊗ · · · ⊗ gv d S d ∋ σ : ( v 1 ⊗ · · · ⊗ v d ) σ = v σ (1) ⊗ · · · ⊗ v σ ( d ) Extending the actions to the group algebras we obtain the algebra homomorphisms � V ⊗ d � ρ : RGL n ( R ) → End RS d � V ⊗ d � ψ : RS d → End RGL n ( R ) . � V ⊗ d � The endomorphism algebra S R ( n , d ) = End RS d is called Schur algebra . We say that (classical) Schur–Weyl duality holds for a ring R if the maps ρ and ψ are surjective. 2

When does Classical Schur–Weyl duality hold? Theorem If ρ : RGL n ( R ) → S R ( n , d ) is surjective then � V ⊗ d � ψ : RS d → End RGL n ( R ) is surjective. 3

When does Classical Schur–Weyl duality hold? Theorem If ρ : RGL n ( R ) → S R ( n , d ) is surjective then � V ⊗ d � ψ : RS d → End RGL n ( R ) is surjective. R ∗ - set of all units of the commutative ring R ; 3

When does Classical Schur–Weyl duality hold? Theorem If ρ : RGL n ( R ) → S R ( n , d ) is surjective then � V ⊗ d � ψ : RS d → End RGL n ( R ) is surjective. R ∗ - set of all units of the commutative ring R ; Theorem ([4, Theorem 3.6.]) Let n , d ∈ N be natural numbers. Let R be a commutative ring with identity that contains a set S which has the following properties: ∀ x , y ∈ S , x � = y = ⇒ x − y ∈ R ∗ ; | S | > d. Then the algebra homomorphism ρ : RGL n ( R ) → S R ( n , d ) is surjective, that is, Schur–Weyl duality holds. 3

When does Classical Schur–Weyl duality hold? Theorem If ρ : RGL n ( R ) → S R ( n , d ) is surjective then � V ⊗ d � ψ : RS d → End RGL n ( R ) is surjective. R ∗ - set of all units of the commutative ring R ; Theorem ([4, Theorem 3.6.]) Let n , d ∈ N be natural numbers. Let R be a commutative ring with identity that contains a set S which has the following properties: ∀ x , y ∈ S , x � = y = ⇒ x − y ∈ R ∗ ; | S | > d. Then the algebra homomorphism ρ : RGL n ( R ) → S R ( n , d ) is surjective, that is, Schur–Weyl duality holds. ρ : RGL n ( R ) → S R ( n , d ) is not surjective F 2 , n = d = 2. 3

Schur–Weyl duality between S R ( n , d ) and S d S R ( n , d ) V ⊗ d S d 4

Schur–Weyl duality between S R ( n , d ) and S d S R ( n , d ) V ⊗ d S d � V ⊗ d � End RS d = S R ( n , d ) ∋ η : η · ( v 1 ⊗ · · · ⊗ v d ) = η ( v 1 ⊗ · · · ⊗ v d ) 4

Schur–Weyl duality between S R ( n , d ) and S d S R ( n , d ) V ⊗ d S d � V ⊗ d � End RS d = S R ( n , d ) ∋ η : η · ( v 1 ⊗ · · · ⊗ v d ) = η ( v 1 ⊗ · · · ⊗ v d ) There is an algebra homomorphism � V ⊗ d � ψ : RS d → End S R ( n , d ) . 4

Schur–Weyl duality between S R ( n , d ) and S d S R ( n , d ) V ⊗ d S d � V ⊗ d � End RS d = S R ( n , d ) ∋ η : η · ( v 1 ⊗ · · · ⊗ v d ) = η ( v 1 ⊗ · · · ⊗ v d ) There is an algebra homomorphism � V ⊗ d � ψ : RS d → End S R ( n , d ) . Theorem ([4, Corollary 3.5.]) ψ is surjective for every commutative ring R. If n ≥ d, then ψ is an isomorphism. 4

Strength of Schur–Weyl duality R - noetherian ring D = Hom R ( − , R ) - standard duality 5

Strength of Schur–Weyl duality R - noetherian ring D = Hom R ( − , R ) - standard duality The homomorphism ρ : RGL n ( R ) → S R ( n , d ) induces by restriction of scalars the functor H R : S R ( n , d )-mod → ( RGL n ( R )) d -mod 5

Strength of Schur–Weyl duality R - noetherian ring D = Hom R ( − , R ) - standard duality The homomorphism ρ : RGL n ( R ) → S R ( n , d ) induces by restriction of scalars the functor H R : S R ( n , d )-mod → ( RGL n ( R )) d -mod ⇒ H R equivalence. SW duality on R = 5

Strength of Schur–Weyl duality R - noetherian ring D = Hom R ( − , R ) - standard duality The homomorphism ρ : RGL n ( R ) → S R ( n , d ) induces by restriction of scalars the functor H R : S R ( n , d )-mod → ( RGL n ( R )) d -mod ⇒ H R equivalence. SW duality on R = ψ is related to the Schur functor F R = Hom S R ( n , d ) ( V ⊗ d , − ): S R ( n , d )-mod → RS d -mod , n ≥ d . 5

Strength of Schur–Weyl duality R - noetherian ring D = Hom R ( − , R ) - standard duality H R : S R ( n , d )-mod → ( RGL n ( R )) d -mod ⇒ H R equivalence. SW duality on R = ψ is related to the Schur functor F R = Hom S R ( n , d ) ( V ⊗ d , − ): S R ( n , d )-mod → RS d -mod , n ≥ d . S R ( n , d )-proj : S R ( n , d )-proj → add RSd DV ⊗ d ⇒ F R ψ isomorphism = equivalence. 5

Strength of Schur–Weyl duality R - noetherian ring D = Hom R ( − , R ) - standard duality H R : S R ( n , d )-mod → ( RGL n ( R )) d -mod ⇒ H R equivalence. SW duality on R = ψ is related to the Schur functor F R = Hom S R ( n , d ) ( V ⊗ d , − ): S R ( n , d )-mod → RS d -mod , n ≥ d . S R ( n , d )-proj : S R ( n , d )-proj → add RSd DV ⊗ d ⇒ F R ψ isomorphism = equivalence. The functor G R = Hom RSd ( DV ⊗ d , − ) is right adjoint to F R . The strength of SW duality on S R ( n , d ) is measured by the degree to which G R fails to be exact on RS d -mod. 5

Strength of Schur–Weyl duality R - noetherian ring D = Hom R ( − , R ) - standard duality H R : S R ( n , d )-mod → ( RGL n ( R )) d -mod ⇒ H R equivalence. SW duality on R = F R = Hom S R ( n , d ) ( V ⊗ d , − ): S R ( n , d )-mod → RS d -mod , n ≥ d . S R ( n , d )-proj : S R ( n , d )-proj → add RSd DV ⊗ d ⇒ F R ψ isomorphism = equivalence. The functor G R = Hom RSd ( DV ⊗ d , − ) is right adjoint to F R . The strength of SW duality on S R ( n , d ) is measured by the degree to which G R fails to be exact on RS d -mod. Theorem ([6, Theorem 5.1], C.) add Z Sd DV ⊗ d is exact. G F 2 G Z add F 2 Sd DV ⊗ d is not exact. 5

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