  Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$. Its Weyl
group is the symmetric group $S_n$. Let $L_w$ be the highest weight module with
highest weight $-w\rho-\rho$. It is well-known that its associated variety
$V(L_w)$ is constant on each Kazhdan-Lusztig right cell. Borho-Brylinski proved
that $V(L_w)$ is irreducible if the Schubert variety $X_w$ is smooth. In this
paper, we want to describe some Young tableaux (KL right cells) containing only
smooth elements or non-smooth elements. This result may provide some examples
of highest weight modules $L_w$ with irreducible or reducible associated
varieties.
