Abstract
The classical case of Schur–Weyl duality states that the actions of the group algebras of GLn and Sd on the dth-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 and for any finite field when d is sufficiently large.
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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.