  This work is part of a project on weight bases for the irreducible
representations of semisimple Lie algebras with respect to which the
representation matrices of the Chevalley generators are given by explicit
formulas. In the case of sl_n, the celebrated Gelfand-Tsetlin basis is the only
such basis known. Using the setup of supporting graphs developed by Donnelly,
we present a simple combinatorial proof of the Gelfand-Tsetlin formulas based
on a rational function identity. Some properties of the Gelfand-Tsetlin basis
are derived via an algorithm for solving certain equations on the lattice of
semistandard Young tableaux.
