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

TL;DR

This paper extends the Gelfand--Tsetlin basis construction from finite-dimensional to infinite-dimensional irreducible representations of the general linear group, providing explicit formulas and parameterizations.

Contribution

It introduces the concept of infinite Gelfand--Tsetlin schemes and constructs an explicit Gelfand--Tsetlin basis for infinite-dimensional irreducible representations.

Findings

Defined infinite Gelfand--Tsetlin schemes for infinite-dimensional groups

Constructed explicit Gelfand--Tsetlin basis using colimits of representations

Extended finite-dimensional basis concepts to infinite-dimensional setting

Abstract

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.