  Let $\mathfrak{g}$ be a simple complex Lie algebra with a Cartan subalgebra
$\mathfrak{h}$. We fix a standard parabolic subalgebra $\mathfrak{p}\supset
\mathfrak{h}$. The socular simple modules play an important role in the
parabolic versions of category $\mathcal{O}^{\mathfrak{p}}$. From Irving's
work, we know that these modules are just those modules with largest possible
Gelfand-Kirillov dimension in $\mathcal{O}^{\mathfrak{p}}$. In this article, we
will give an explicit characterization for these modules of
$\mathfrak{sl}(n,\mathbb{C})$. Our characterization is given in the information
of the corresponding highest weight and Young tableau.
