  We show that a specialization in Weyl character formula can be carried out in
such a way that its right-hand side becomes simply a Schur Function. For this,
we need the use of fundamental weights.
  In the generic definition, an Elementary Schur Function $S_Q(x_1,x_2,..,x_Q)$
of degree Q is known to be defined by some polynomial of Q indeterminates $
x_1,x_2,..,x_Q $. It is also known that definition of Elementary Schur
Functions can be generalized in such a way that for any partition $(Q_k)$ of
weight Q and length k one has a Generalized Schur Function
$S_{(Q_k)}(x_1,x_2,..,x_Q)$. When they are considered for $A_{N-1}$ Lie
algebras, a kind of degeneration occurs for these generic definitions. This is
mainly due to the fact that, for an $A_{N-1}$ Lie algebra, only a finite number
of indeterminates, namely (N-1), can be independent. This leads us to define
{\bf Degenerated Schur Functions} by taking, for $Q > N-1$, all the
indeterminates $x_Q$ to be non-linearly dependent on first (N-1) indeterminates
$x_1,x_2,..,x_{N-1}$. With this in mind, we show that for each and every
dominant weight of $A_{N-1}$ we always have a (Degenerated) Schur Function
which provides the right-hand side of Weyl character formula.
  Generalized Schur Functions are known to be expressed by determinants of some
matrices of Elementary Schur Functions. We would like to call these expressions
{\bf multiplicity rules}. This is mainly due to the fact that, to calculate
weight multiplicities, these rules give us an efficient method which works
equally well no matter how big is the rank of algebras or the dimensions of
representations.
