Combinatorial construction of Gelfand-Tsetlin modules for $\mathfrak{gl}_n$

We propose a new effective method of constructing explicitly Gelfand -Tsetlin modules for $\mathfrak{gl}_n$. We obtain a large family of simple modules that have a basis consisting of Gelfand-Tsetlin tableaux, the action of the Lie algebra is given by the Gelfand-Tsetlin formulas and with all Gelfand-Tsetlin multiplicities equal $1$. As an application of our construction we prove necessary and sufficient condition for the Gelfand and Graev's continuation construction to define a module which was conjectured by Lemire and Patera.