Gelfand–Tsetlin basis for irreducible representations of the infinite-dimensional general linear group

We consider the problem of constructing a Gelfand–Tsetlin basis in irreducible representations of the infinite-dimensional general linear group. For finite-dimensional irreducible representations of a general linear group, all elements of the Gelfand–Tsetlin basis are parameterized by Gelfand–Tsetlin schemes. We extend this definition to infinite Gelfand–Tsetlin schemes, which, in turn, parameterize elements of the Gelfand–Tsetlin basis of an irreducible representation of the infinite-dimensional complete linear group. Using properties of co-limits of representations with the highest weight, we present an explicit form of the Gelfand–Tsetlin basis.