Combinatorial Constructions of Weight Bases: The Gelfand-Tsetlin Basis
  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. 
Patricia Hersh and Cristian Lenart