The explicit construction of irreducible representations of the quantum algebras U[sub q](sl(n)).

The duality between the quantum algebra U[SUBq](sl(n)) and the Hecke algebra H[SUBm](q[SUP2]) first pointed out by Jimbo is exploited to construct explicit irreducible representations of U[SUBq](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[SUBq](sl(3)) are given. The complications that arise in the non-generic case for which q is a primitive p-th root of unity are then addressed. Explicit results on both irreducible and indecomposable representations are presented. [ABSTRACT FROM AUTHOR]