Classically, the symmetric groups were used to describe representations of the general linear group Gl(N) by decomposing tensor products of its vector representations. For other Lie types this approach becomes much more complicated or is not available. For quantum groups, the symmetric groups are replaced by braid groups which have a far richer representation theory. The purpose of this article is to describe how braid groups can be used to decompose tensor products of representations of quantum groups and the corresponding objects in fusion categories.

In Section 1 we review Schur duality, and its analog for quantum groups of type A discovered by Jimbo. Using representations of Hecke algebras of type A, one can completely classify any rigid semisimple monoidal tensor category whose Grothendieck semiring coincides with the one of Rep(SU(N)). This result also extends to the associated fusion categories.

In Section 2, braid representations are considered which can be regarded as a q-analog of Brauer's centralizer algebra for orthogonal and symplectic groups.