W-graph determining elements in type A

Van Minh Nguyen

Abstract

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.

Keywords: Coxeter group, W-graph, Kazhdan–Lusztig cell, skew diagram, standard tableau.

AMS Subject Classification: Primary 20C08; secondary 20.85.