Schur–Weyl duality over commutative rings
T Cruz - Communications in Algebra, 2019 - Taylor & Francis
Communications in Algebra, 2019•Taylor & Francis
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 Z and for any finite field
when d is sufficiently large.
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 Z and for any finite field
when d is sufficiently large.
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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