# Classification of irreducible Gelfand-Tsetlin modules of sl(3)

## TL;DR

This paper classifies and explicitly constructs all irreducible Gelfand-Tsetlin modules of sl(3), detailing their realizations and structures, including those with infinite-dimensional weight spaces, using tableaux and subquotients.

## Contribution

It provides the first complete classification and explicit realization of all irreducible Gelfand-Tsetlin modules of sl(3), including those with infinite-dimensional weight spaces.

## Key findings

- All simple Gelfand-Tsetlin sl(3)-modules with infinite-dimensional weight spaces are listed.
- Modules are realized using regular and derivative Gelfand-Tsetlin tableaux.
- Simple modules are expressed as subquotients of localized Gelfand-Tsetlin E_{21}-injective modules.

## Abstract

We provide a classification and an explicit realization of all irreducible Gelfand-Tsetlin modules of the complex Lie algebra sl(3). The realization of these modules uses regular and derivative Gelfand-Tsetlin tableaux. In particular, we list all simple Gelfand-Tsetlin sl(3)-modules with infinite-dimensional weight spaces. Also, we express all simple Gelfand-Tsetlin sl(3)-modules as subquotionets of localized Gelfand-Tsetlin E_{21}-injective modules.