The Gel'fand-Tsetlin Selection Rules and Representations of Quantum   Algebras
  The problem of construction of irreducible representations of quantum $A^q_n$ algebras is solved at the level of explicit integration of the linear (inhomogeneous) system in finite differences in the n-dimensional space. The general solution of this system is given explicitly and particular ones, which correspond to the irreducible representations are selected. 
A. N. Leznov