Schur–Weyl duality over commutative rings

Abstract

The classical case of Schur-Weyl duality states that the actions of the group algebras of GLn and S d on the d th-tensor power of a free module of finite rank centralize each other. We show that Schur-Weyl duality holds for commutative rings where enough scalars can be chosen whose non-zero differences are invertible. This implies all the known cases of Schur-Weyl duality so far. We also show that Schur-Weyl duality fails for Z and for any finite field when d is sufficiently large. 1 Introduction Schur-Weyl duality is a connection between the general linear group and the symmetric group. More specifically, consider n, d ∈ N and let V = R n be the free module of rank n over a commutative ring with identity R. The symmetric group S d acts on the d th-tensor power, V ⊗ d = V ⊗ R • • • ⊗ R V , of the module V by place permutation, that is, σ(v 1 ⊗ • • • ⊗ v d) = v σ −1 (1) ⊗ • • • ⊗ v σ −1 (d) , σ ∈ S d , v i ∈ V. Definition 1.1. [Gre80] The subalgebra End RS d V ⊗ d of the endomorphism algebra End R V ⊗ d is called the Schur algebra. We will denote it by S R (n, d). On the other hand, the general linear group acts on V by multiplication, and thus on the tensor product V ⊗ d by the diagonal action, that is, g(v 1 ⊗ • • • ⊗ v d) = gv 1 ⊗ • • • ⊗ gv d , g ∈ GL n (R), v i ∈ V. These two actions commute, so, by extending these actions to the group algebras, one gets two natural homomorphisms: ρ : RGL n (R) → S R (n, d), ψ : RS d → End RGLn(R) V ⊗ d. Definition 1.2. We say that Schur-Weyl duality holds if the two algebra homomorphisms ρ and ψ are surjective.

Perturbative 4D conformal field theories and representation theory of diagram algebras

“…In other words we should treat these as scalars. Schur-Weyl duality in the more general setting of rings over modules is discussed in [51].…”

A Characterisation of Morita Algebras in Terms of Covers

“…The proof of Theorem 1 involves the study of double centralizer properties and a reformulation of the definition of Morita algebras using the Nakayama functor. Prominent examples of double centralizer properties are Soergel's double centralizer theorem [12], classical Schur-Weyl duality [6] and its many generalizations (see for example [2]).…”

A characterisation of Morita algebras in terms of covers

“…The proof of Theorem 1 involves the study of double centralizer properties and a reformulation of the definition of Morita algebras using the Nakayama functor. Prominent examples of double centralizer properties are Soergel's double centralizer theorem [Soe90], classical Schur-Weyl duality [Gre80] and its many generalizations (see for example [Cru19]).…”