# The Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases

## TL;DR

This paper extends the Gelfand-Tsetlin construction for simple modules of $sl_n$, providing new theorems, proofs, and explicit monomial bases to deepen understanding of their structure.

## Contribution

It introduces new theorems and proofs to explicitly construct monomial bases for simple $sl_n$ modules, advancing the classical Gelfand-Tsetlin theory.

## Key findings

- Explicit monomial bases for simple $sl_n$ modules are constructed.
- Theoretical framework for Gelfand-Tsetlin realization is extended.
- Provides rigorous proofs for the new basis constructions.

## Abstract

The most famous simple Lie algebra is $sl_n$ (the $n \times n$ matrices with trace equals $0$). The representation theory for $sl_n$ has been one of the most important research areas for the past hundred years and within their the simple finite-dimensional modules have become very important. They are classified and Gelfand and Tsetlin actually gave an explicit construction of a basis for every simple module. We extend it by providing theorems and proofs, and construct monomial bases of the simple module.