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Given a weight of $sl(n,\mbb{C})$, we derive a system of variable-coefficientsecond-order linear partial differential equations that determines the singularvectors in the corresponding Verma module, and a differential-operatorrepresentation of the symmetric group $S_n$ on the related space of truncatedpower series. We prove that the solution space of the system of partialdifferential 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 givenby those $\sgm(1)$ that are polynomials. The well-known results of Verma,Bernstein-Gel'fand-Gel'fand and Jantzen for the case of $sl(n,\mbb{C})$ arenaturally included in our almost elementary approach of partial differentialequations.

Differential-Operator Representations of $S_n$ and Singular Vectors in Verma Modules