Combinatorial construction of Gelfand–Tsetlin modules for gln

Abstract We propose a new effective method of constructing explicitly Gelfand–Tsetlin modules for 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.