Kazhdan-Lusztig left cell preorder and dominance order

Abstract

Let "${\le }_L$" 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 $x,y\in S_n$, $x{\le }_{L}y$ only if $Q(y)⊵Q(x)$, where "$⊵$" 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 $C_w^{\prime }$ can be expressed as a linear combination of some $m_{uv}$ which satisfies that $u⊵P(w{)}^{\ast }$, $v⊵Q(w{)}^{\ast }$, where $t^{\ast }$ denotes the conjugate of $t$ for each standard tableau $t$, $\{m_{st}\mid s,t\in Std(\lambda ),\lambda ⊢n\}$ is the Murphy basis of the Iwahori-Hecke algebra $H_v(S_n)$ associated to $S_n$.