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 $\mathrm{S}\mathrm{y}\mathrm{m}(n)$ for some positive integer $n$ and $S$ with the set of simple transpositions ${,(i,i+1)\mid 1\leqslant i\leqslant n-1,}$\mathrm{.}Let$\leqslant_{\mathsf L}$denotetheleftweakorderon$W$,andforeach$J\subseteq S$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 $\{x\in W\mid w_J{⩽}_{\mathsf{L}}x{⩽}_{\mathsf{L}}w{w}_J\}$ is a nonempty union of Kazhdan-Lusztig left cells. These are also the pairs $(w,J)$ such that $\mathcal{I}(w)=\{v\in W\mid v{⩽}_{\mathsf{L}}w\}$ is a $W\text{\hspace{0.17em}⁣}$-graph ideal with respect to $J$. Moreover, for each such pair the elements of $\mathcal{I}(w)$ are in bijective correspondence with the standard tableaux associated with the corresponding skew diagram.