Kazhdan-Lusztig left cell preorder and dominance order

TL;DR

This paper establishes a connection between the Kazhdan-Lusztig left cell preorder on symmetric groups and the dominance order on standard tableaux, extending previous results and providing new algebraic expressions for basis elements.

Contribution

It proves that the Kazhdan-Lusztig preorder implies a dominance order relation on tableaux and generalizes earlier results on basis element expressions in the Iwahori-Hecke algebra.

Findings

The preorder $\leq_L$ implies $Q(y) \unrhd Q(x)$ for tableaux.

Each Kazhdan-Lusztig basis element can be expressed via Murphy basis elements with dominance conditions.

Generalization of Geck's result on basis element expansions.

Abstract

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 $S_n$.