  Let "$\leq_L$" be the Kazhdan-Lusztig left cell preorder on the symmetric
group $S_n$. Let $w\mapsto (P(w),Q(w))$ be the Robinson-Schensted-Knuth
correspondence between $S_n$ and the set of standard tableaux with the same
shapes. We prove that for any $x,y\in S_n$, $x\leq_L y$ only if $Q(y)\unrhd
Q(x)$, where "$\unrhd$" is the dominance (partial) order between standard
tableaux. As a byproduct, we generalize an earlier result of Geck by showing
that each Kazhdan-Lusztig basis element $C'_w$ can be expressed as a linear
combination of some $m_{uv}$ which satisfies that $u\unrhd P(w)^*$, $v\unrhd
Q(w)^*$, where $t^*$ denotes the conjugate of $t$ for each standard tableau
$t$, $\{m_{st} \mid s,t\in Std(\lambda),\lambda\vdash n\}$ is the Murphy basis
of the Iwahori-Hecke algebra $H_{v}(S_n)$ associated to $S_n$.
