Schur--Weyl duality over commutative rings

TL;DR

This paper extends Schur--Weyl duality to certain commutative rings with invertible differences, unifying known cases, but demonstrates failure over integers and finite fields for large tensor powers.

Contribution

It generalizes Schur--Weyl duality to a broad class of commutative rings and identifies conditions under which it fails over integers and finite fields.

Findings

Schur--Weyl duality holds over rings with invertible scalar differences.

Duality fails over or large tensor powers.

Unifies known cases of Schur--Weyl duality.

Abstract

The classical case of Schur--Weyl duality states that the actions of the group algebras of $GL_n$ 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 $\mathbb{Z}$ and for any finite field when $d$ is sufficiently large.