Singular Gelfand Tsetlin modules of gl(n)
Abstract The classical Gelfand Tsetlin formulas provide a basis in terms of tableaux and an explicit action of the generators of gl n for every irreducible finite dimensional gl n module. These formulas can be used to define a gl n module structure on some infinite dimensional modules the so called generic Gelfand Tsetlin modules. The generic Gelfand Tsetlin modules are convenient to work with since for every generic tableau there exists a unique irreducible generic Gelfand Tsetlin module containing this tableau as a basis element. In this paper we initiate the systematic study of a large class of non generic Gelfand Tsetlin modules the class of 1 singular Gelfand Tsetlin modules. An explicit tableaux realization and the action of gl n on these modules is provided using a new construction which we call derivative tableaux. Our construction of 1 singular modules provides a large family of new irreducible Gelfand Tsetlin modules of gl n and is a part of the classification of all such irreducible modules for n 3