  Given a weight of $sl(n,\mbb{C})$, we derive a system of variable-coefficient
second-order linear partial differential equations that determines the singular
vectors in the corresponding Verma module, and a differential-operator
representation of the symmetric group $S_n$ on the related space of truncated
power series. We prove that the solution space of the system of partial
differential equations is exactly spanned by $\{\sgm(1)\mid \sgm\in S_n\}$.
Moreover, the singular vectors of $sl(n,\mbb{C})$ in the Verma module are given
by those $\sgm(1)$ that are polynomials. The well-known results of Verma,
Bernstein-Gel'fand-Gel'fand and Jantzen for the case of $sl(n,\mbb{C})$ are
naturally included in our almost elementary approach of partial differential
equations.
