The duality between the quantum algebra U_q(sl(n)) and the Hecke algebra H_m(q~2) first pointed out by Jimbo is exploited to construct explicit irreducible representations of U_q(sl(n)). The method is based on the use of Young tableaux and involves the notion of q-dependent Young symmetrisers. A key role is played by q-dependent generalisations of the Garnir identities. The appropriate algorithm is first described and illustrated in the generic case for which q is not a root of unity. All matrix elements for the irreducible representations of U_q(sl(3)) are given. The complications that arise in the non-generic case for which q is a primitive p-th root on unity are then addressed. Explicit results on both irreducible and indecomposable representations are presented.
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