## 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.

: Primary 20C08; secondary 20.85.

This paper is available as a pdf (232kB) file.

 Tuesday, March 3, 2015