The Gelfand-Tsetlin basis for irreducible representations of an infinite-dimensional general linear group

We consider the problem of constructing a Gelfand–Tsetlin basis in irreducible representations of an infinite-dimensional general linear group. For a finite-dimensional irreducible representation 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 an infinite-dimensional complete linear group. Using the properties of colimits of representations with the highest weight, we present an explicit form of the Gelfand–Tsetlin basis.