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**Quantized mixed tensor space and Schur-Weyl duality.**
*(English)*
Zbl 1290.17012

Classically, the Schur-Weyl duality corresponds to the statement that the action of \(\mathfrak{gl}(n)\) and that of the symmetric group \(S_m\) satisfy the double centralizer property on the \(m\)th tensor power of the tautological representation of \(\mathfrak{gl}(n)\).

This theorem has been generalized to many different settings, such as to the tensor product of tensor powers of the tautological module and its dual. In this setting the symmetric group needs to be replaced by a walled Brauer algebra. This has in turn be generalized to generic quantum group versions of \(\mathfrak{gl}(n)\).

In the paper under review, the authors prove the Schur-Weyl duality for quantum group versions of \(\mathfrak{gl}(n)\) in a very general setting. The corresponding walled Brauer algebra, has been introduced by R. Leduc. One direction of the theorem, that the centralizer of the quantum group is generated by the Brauer algebra, was already proved by the same authors in an earlier paper.

This theorem has been generalized to many different settings, such as to the tensor product of tensor powers of the tautological module and its dual. In this setting the symmetric group needs to be replaced by a walled Brauer algebra. This has in turn be generalized to generic quantum group versions of \(\mathfrak{gl}(n)\).

In the paper under review, the authors prove the Schur-Weyl duality for quantum group versions of \(\mathfrak{gl}(n)\) in a very general setting. The corresponding walled Brauer algebra, has been introduced by R. Leduc. One direction of the theorem, that the centralizer of the quantum group is generated by the Brauer algebra, was already proved by the same authors in an earlier paper.

Reviewer: Kevin Coulembier (Gent)