W-graph determining elements in type A

Let $(W,S)$ be a Coxeter system of type $A$, so that $W$ can be identified with the symmetric group $\mathrm{Sym}(n)$ for some positive integer $n$ and $S$ with the set of simple transpositions $\{\,(i,i+1)\mid 1\leqslant i\leqslant n-1\,\}$. Let $\leqslant_{\mathsf L}$ denote the left weak order on $W$, and for each $J\subseteq S$ let $w_J$ be the longest element of the subgroup $W_J$ generated by $J$. We show that the basic skew diagrams with $n$ boxes are in bijective correspondence with the pairs $(w,J)$ such that the set $\{\,x\in W\mid w_J\leqslant_{\mathsf L} x\leqslant_{\mathsf L} ww_J\,\}$ is a nonempty union of Kazhdan-Lusztig left cells. These are also the pairs $(w,J)$ such that $\mathscr{I}(w)=\{\,v\in W\mid v\leqslant_{\mathsf L} w\,\}$ is a $W\!$-graph ideal with respect to $J$. Moreover, for each such pair the elements of $\mathscr{I}(w)$ are in bijective correspondence with the standard tableaux associated with the corresponding skew diagram.

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