The Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases

The most famous simple Lie algebra is $s{l}_n$ (the $n\times n$ matrices with trace equals $0$). The representation theory for $s{l}_n$ 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.