Schur–Weyl duality over commutative rings

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.

Keywords:

• Schur–Weyl duality
• Schur algebra

2010 Mathematics Subject Classification:

• 20C30
• 20G05
• 20G43

Acknowledgments

All these results are part of my master’s thesis Schur–Weyl duality written in Portuguese and defended in June 2017 at the University of Coimbra. I would like to thank my master’s thesis advisors Doctor Ana Paula Santana and Doctor Ivan Yudin for all the guidance provided and their valuable comments on this work.

Disclosure statement

No potential conflict of interest was reported by the author.