  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.
