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

E. A. Movchan

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 30, Tome 532 (2024), pp. 235-256

Résumé

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.